Making Inferences & Justifying Conclusions - AP Statistics

Solving questions related to this standard requires an understanding of definitions common to statistics. Specifically, this question is testing your knowledge of the difference between two fundamental statistical concepts: sample statistics and population parameters. Let's begin by discussing the differences between these two measures. Later, we will use this information to solve the problem.

First let's discuss what is meant by the term population. In statistics, a "population" is described as the entire group that is to be studied. An example of a population in the natural sciences would be every giant panda of the species _Ailuropoda melanoleuca_in the wild (1864 individuals according to the World Wildlife Foundation)—not captivity. Now, let's identify what is meant by the term population parameter. A "population parameter" is a statistic that is found by sampling the entire population. For example, the mean weight of the entire wild population of giant pandas in the world would be an example of a population parameter (i.e. the mean weight of all 1864 pandas). Next, we will discuss sample populations and statistics.

A "sample" is the subset of a population that is being studied. For example, researchers for a university want to study giant pandas in the wild but can only access a group of 100 pandas sampled in Sichuan, China. Data collected from this particular study would be known as a sample statistic (e.g. the mean weight of pandas in the Sichuan region). It is important to note that the external validity of some sample statistics are hindered. The external validity of a statistic is its ability to be applied to other samples and remain valid. If locals fed pandas in the Sichuan region, then their mean weight may be greater than those of the southern or northern regions. In this instance, the mean would not be representative of other populations of giant pandas.

Last, we should note that certain sample populations are better than others at predicting population parameters. A population parameter can be considered to be the true statistic of a given population while a sample statistic is only an estimate of a part or subset of the population. Simple random samples are good predictors of population parameters and can be used to estimate them. They are collected when every member in a population has an equal chance of being chosen (e.g. randomly selecting 100 of the 1864 pandas in the world).

Now, let's use this information to solve the question. The designer wants to know the preferences of potential car buyers; however, he only samples three major US cities. The data collected from this survey is an example of a sample statistic. It did not gather information from all of the potential car buyers for the particular company; therefore, the best answer is "sample statistic."

Solving questions related to this standard requires an understanding of definitions common to statistics. Specifically, this question is testing your knowledge of the difference between two fundamental statistical concepts: sample statistics and population parameters. Let's begin by discussing the differences between these two measures. Later, we will use this information to solve the problem.

First let's discuss what is meant by the term population. In statistics, a "population" is described as the entire group that is to be studied. An example of a population in the natural sciences would be every giant panda of the species _Ailuropoda melanoleuca_in the wild (1864 individuals according to the World Wildlife Foundation)—not captivity. Now, let's identify what is meant by the term population parameter. A "population parameter" is a statistic that is found by sampling the entire population. For example, the mean weight of the entire wild population of giant pandas in the world would be an example of a population parameter (i.e. the mean weight of all 1864 pandas). Next, we will discuss sample populations and statistics.

A "sample" is the subset of a population that is being studied. For example, researchers for a university want to study giant pandas in the wild but can only access a group of 100 pandas sampled in Sichuan, China. Data collected from this particular study would be known as a sample statistic (e.g. the mean weight of pandas in the Sichuan region). It is important to note that the external validity of some sample statistics are hindered. The external validity of a statistic is its ability to be applied to other samples and remain valid. If locals fed pandas in the Sichuan region, then their mean weight may be greater than those of the southern or northern regions. In this instance, the mean would not be representative of other populations of giant pandas.

Last, we should note that certain sample populations are better than others at predicting population parameters. A population parameter can be considered to be the true statistic of a given population while a sample statistic is only an estimate of a part or subset of the population. Simple random samples are good predictors of population parameters and can be used to estimate them. They are collected when every member in a population has an equal chance of being chosen (e.g. randomly selecting 100 of the 1864 pandas in the world).

Now, let's use this information to solve the question. The designer wants to know the preferences of potential car buyers; however, he only samples three major US cities. The data collected from this survey is an example of a sample statistic. It did not gather information from all of the potential car buyers for the particular company; therefore, the best answer is "sample statistic."

This question requires us to understand the differences between the following research methods: observational studies, surveys, and experiments. We will do this by identifying the marked characteristics of each research method and understand when each should be used in research.

First, we will discuss observational studies. Observational studies can be described as anthropologic or ethnographic studies. In these studies, researchers observe a particular event and develop conclusions on these groups or phenomena based upon direct observation. For example, researchers have explored tribal groups in many countries as well as social phenomena such as the baby boomer generation as well as deviant groups (e.g. gangs and paramilitary organizations)

Second, surveys are typically composed of questionnaires or interviews. They are used to ask a sample of participants questions in order to gather information that is to be generalized to the greater population. Survey research often employs the use of random sampling techniques in order to get a representative sample of the population. In this way, they have a high level of external validity. In other words, the sample statistics derived from the study may be used to make valid inferences of the population parameters of society at large. An example of survey research could include restaurant questionnaires related to satisfaction as well as the United States Census (it is important to note that a census has high external validity because it attempts to sample every member of a population). Another survey with high external validity could be a survey that takes a simple random sample of high school, college, and secondary school graduates and their occupational wages post graduation. This survey could find a correlation between level of education attained and potential income. It is important to note that correlation does not equal causation. A researcher could find that shark attacks and ice creams sales at the beach are positively correlated; however, this does not mean that shark attacks cause ice cream sales or that ice cream consumption leads to shark attacks. Instead, both of these particular phenomena are caused by a hidden or lurking variable: warm weather and greater numbers of beach goers.

Last, we will discuss experiments. Experiments employ methods of random sampling; furthermore, they use procedures to assign subjects and variables to control or treatment groups. The treatment group receives the treatment in the trial while the control group is given a substitute that would not alter standard conditions. For example, scientists studying the effect of a blood pressure pill on a group of individuals with high blood pressure would give the treatment group the pill and substitute the treatment for a placebo or sugar pill in the control group. Experiments use double blind methodologies to assign subjects to control and experimental groups. These methods randomly assign subjects to either group; furthermore, they hide the assignments from both subjects and researchers. This prevents experimenter biases as well as limits the placebo effect—in which subjects tend to display the symptoms of the treatment group while being part of the control group. These studies collect representative samples and control for extraneous variables; therefore, they have strong external validity. It is important to note that experiments directly manipulate variables and can be used to identify causal relationships; therefore, they can be said to be strong predictors of causality.

Now, let's use this information to answer the question. The characteristics of these research methods have been tabulated in the following table:

Observational studies use purposeful samples not random samples. In other words, they pick a particular group or event and study it directly in order to learn about it with little desire to extend this information to the greater population; therefore, they also lack external validity.

In the problem, researchers used a questionnaire in order to find the preferences and satisfaction of customers at a specialty cheese company. If we look at the characteristics given we can see that they are not assigning a treatment and they are not trying to indicate causality (instead, satisfaction). On the other hand, they are trying to generalize these results to the greater population of cheese consumers (i.e. from the feedback of recent consumers) and they are using a random sample of recent consumers. From this information, we can see that they have no treatment assignment or investigation of causality; furthermore, they want a high external validity and are using methods of random sampling. They are using a survey research design.

This question requires us to understand the differences between the following research methods: observational studies, surveys, and experiments. We will do this by identifying the marked characteristics of each research method and understand when each should be used in research.

First, we will discuss observational studies. Observational studies can be described as anthropologic or ethnographic studies. In these studies, researchers observe a particular event and develop conclusions on these groups or phenomena based upon direct observation. For example, researchers have explored tribal groups in many countries as well as social phenomena such as the baby boomer generation as well as deviant groups (e.g. gangs and paramilitary organizations)

Second, surveys are typically composed of questionnaires or interviews. They are used to ask a sample of participants questions in order to gather information that is to be generalized to the greater population. Survey research often employs the use of random sampling techniques in order to get a representative sample of the population. In this way, they have a high level of external validity. In other words, the sample statistics derived from the study may be used to make valid inferences of the population parameters of society at large. An example of survey research could include restaurant questionnaires related to satisfaction as well as the United States Census (it is important to note that a census has high external validity because it attempts to sample every member of a population). Another survey with high external validity could be a survey that takes a simple random sample of high school, college, and secondary school graduates and their occupational wages post graduation. This survey could find a correlation between level of education attained and potential income. It is important to note that correlation does not equal causation. A researcher could find that shark attacks and ice creams sales at the beach are positively correlated; however, this does not mean that shark attacks cause ice cream sales or that ice cream consumption leads to shark attacks. Instead, both of these particular phenomena are caused by a hidden or lurking variable: warm weather and greater numbers of beach goers.

Last, we will discuss experiments. Experiments employ methods of random sampling; furthermore, they use procedures to assign subjects and variables to control or treatment groups. The treatment group receives the treatment in the trial while the control group is given a substitute that would not alter standard conditions. For example, scientists studying the effect of a blood pressure pill on a group of individuals with high blood pressure would give the treatment group the pill and substitute the treatment for a placebo or sugar pill in the control group. Experiments use double blind methodologies to assign subjects to control and experimental groups. These methods randomly assign subjects to either group; furthermore, they hide the assignments from both subjects and researchers. This prevents experimenter biases as well as limits the placebo effect—in which subjects tend to display the symptoms of the treatment group while being part of the control group. These studies collect representative samples and control for extraneous variables; therefore, they have strong external validity. It is important to note that experiments directly manipulate variables and can be used to identify causal relationships; therefore, they can be said to be strong predictors of causality.

Now, let's use this information to answer the question. The characteristics of these research methods have been tabulated in the following table:

Observational studies use purposeful samples not random samples. In other words, they pick a particular group or event and study it directly in order to learn about it with little desire to extend this information to the greater population; therefore, they also lack external validity.

In the problem, researchers used a questionnaire in order to find the preferences and satisfaction of customers at a specialty cheese company. If we look at the characteristics given we can see that they are not assigning a treatment and they are not trying to indicate causality (instead, satisfaction). On the other hand, they are trying to generalize these results to the greater population of cheese consumers (i.e. from the feedback of recent consumers) and they are using a random sample of recent consumers. From this information, we can see that they have no treatment assignment or investigation of causality; furthermore, they want a high external validity and are using methods of random sampling. They are using a survey research design.

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are "if/then" statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes three times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice three times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: "Joe has tricked her by using loaded dice."

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are "if/then" statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes three times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice three times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: "Joe has tricked her by using loaded dice."

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes four times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice four times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes five times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice five times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes six times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice six times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes seven times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice seven times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes eight times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice eight times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes nine times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice nine times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes ten times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice ten times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes eleven times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice eleven times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes twelve times in a row is very low. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice twelve times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe has tricked her by using loaded dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes one time in a row is low but somewhat likely. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice three times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe is using fair dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

From this information, she realized that the probability of rolling snake eyes two times in a row is low but somewhat likely. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice three times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe is using fair dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

In order to solve this problem, let's first observe the steps associated with the scientific method. In this particular experiment we can see that the scientists have interest in a particular biological phenomenon: the effects of hormones on plant growth and germination. They develop a hypothesis based on these observations. A hypothesis is a tentative explanation for an observed phenomenon. In this experiment the scientists developed the following hypothesis: abscisic acid will inhibit plant growth, while gibberellins will counteract this germination inhibition. A null hypothesis is a statement of no difference. In other words, plant hormones will have no effect on plant growth.

In order to support or refute this hypothesis, the scientists developed a simulation or experiment. In this scenario, they exposed corn seeds to differing hormone treatments:

1. No treatment
2. Abscisic acid
3. Gibberellins and abscisic acid

They monitored the seeds for fourteen days and created a box plot using the data.

Let's look at how box plots are used to analyze data.

Box plots are broken into five main parts that create four primary regions. Each region contains a quarter of the data in the plot. The plot is initially broken into two regions using the following values: the minimum value, the maximum value, and the median or average of the set. Next, the minimum and median values are divided by the median of the lower half of the data or the first/lower quartile. Likewise, the median and the maximum values are divided by the median of the upper half of the data set or third/upper quartile. If the box is bigger, then there is a greater variance or spread in the data. Also, if the whiskers are very long, then the data possesses outliers.

When data is analyzed in a box plot or chart, it can then be used to make conclusions on the experiment. These conclusions will either support or refute a hypothesis. It is important to note that hypothesis cannot be proven support can only be added for them or against them. Even scientific theories such as gravity are not proven: they are just supported by a wealth of experiments and knowledge.

This brings us to the basis of this question. We need to make a conclusion based on the results of a simulation. Initially, we can see that the treatments had some effect due to the differences between the abscisic acid plot and the control plot; therefore, any choice that said there was not an effect can be disregarded. In the graph, we can clearly see that abscisic acid inhibits plant growth compared to the control; therefore, we can say, "abscisic acid inhibits plant growth." When we compare the control group with the treatment group that received both gibberellins and abscisic acid, we can see that they are comparable. This means that we can infer the following: "gibberellins can counteract abscisic acid inhibition." The correct choice is: "Yes, abscisic acid inhibits plant growth and gibberellins can counteract growth inhibition."

In order to solve this problem, let's first observe the steps associated with the scientific method. In this particular experiment we can see that the scientists have interest in a particular biological phenomenon: the effects of hormones on plant growth and germination. They develop a hypothesis based on these observations. A hypothesis is a tentative explanation for an observed phenomenon. In this experiment the scientists developed the following hypothesis: abscisic acid will inhibit plant growth, while gibberellins will counteract this germination inhibition. A null hypothesis is a statement of no difference. In other words, plant hormones will have no effect on plant growth.

In order to support or refute this hypothesis, the scientists developed a simulation or experiment. In this scenario, they exposed corn seeds to differing hormone treatments:

1. No treatment
2. Abscisic acid
3. Gibberellins and abscisic acid

They monitored the seeds for fourteen days and created a box plot using the data.

Let's look at how box plots are used to analyze data.

Box plots are broken into five main parts that create four primary regions. Each region contains a quarter of the data in the plot. The plot is initially broken into two regions using the following values: the minimum value, the maximum value, and the median or average of the set. Next, the minimum and median values are divided by the median of the lower half of the data or the first/lower quartile. Likewise, the median and the maximum values are divided by the median of the upper half of the data set or third/upper quartile. If the box is bigger, then there is a greater variance or spread in the data. Also, if the whiskers are very long, then the data possesses outliers.

When data is analyzed in a box plot or chart, it can then be used to make conclusions on the experiment. These conclusions will either support or refute a hypothesis. It is important to note that hypothesis cannot be proven support can only be added for them or against them. Even scientific theories such as gravity are not proven: they are just supported by a wealth of experiments and knowledge.

This brings us to the basis of this question. We need to make a conclusion based on the results of a simulation. Initially, we can see that the treatments had some effect due to the differences between the abscisic acid plot and the control plot; therefore, any choice that said there was not an effect can be disregarded. In the graph, we can clearly see that abscisic acid inhibits plant growth compared to the control; therefore, we can say, "abscisic acid inhibits plant growth." When we compare the control group with the treatment group that received both gibberellins and abscisic acid, we can see that they are comparable. This means that we can infer the following: "gibberellins can counteract abscisic acid inhibition." The correct choice is: "Yes, abscisic acid inhibits plant growth and gibberellins can counteract growth inhibition."

In order to solve this question, we need to learn how to infer population parameters from sample statistics. We will do this by reviewing the definitions of parameters and statistics, recalling the processes associated with the scientific method, and by interrogating the information in the problem.

First let's discuss what is meant by the term population. In statistics, a "population" is described as the entire group that is to be studied. An example of a population in the natural sciences would be every giant panda of the species Ailuropoda melanoleuca in the wild (1864 individuals according to the World Wildlife Foundation)—not captivity. Now, let's identify what is meant by the term population parameter. A "population parameter" is a statistic that is found by sampling the entire population. For example, the mean weight of the entire wild population of giant pandas in the world would be an example of a population parameter (i.e. the mean weight of all 1864 pandas). Next, we will discuss sample populations and statistics.

A "sample" is the subset of a population that is being studied. For example, researchers for a university want to study giant pandas in the wild but can only access a group of 100 pandas sampled in Sichuan, China. Data collected from this particular study would be known as a sample statistic (e.g. the mean weight of pandas in the Sichuan region). It is important to note that the external validity of some sample statistics are hindered. The external validity of a statistic is its ability to be applied to other samples and remain valid. If locals fed pandas in the Sichuan region, then their mean weight may be greater than those of the southern or northern regions. In this instance, the mean would not be representative of other populations of giant pandas.

Last, we should note that certain sample populations are better than others at predicting population parameters. A population parameter can be considered to be the true statistic of a given population while a sample statistic is only an estimate of a part or subset of the population. Simple random samples are good predictors of population parameters and can be used to estimate them. They are collected when every member in a population has an equal chance of being chosen (e.g. randomly selecting 100 of the 1864 pandas in the world).

Next, we shall observe the steps associated with the scientific method.

In this method, researchers observe a phenomenon and develop a tentative explanation for it. Afterwards, they develop an experiment, which assigns subjects or variables to control and treatment groups. Experiments measure the change that an independent variable (i.e. treatment) has on a dependent variable (i.e. subject or phenomena under investigation). Scientists repeat or replicate these experiments multiple times in order gather data for statistical analysis. Scientists analyze the data and then use this evidence to either support or refute a hypothesis. Hypotheses can only be supported or refuted: never proven. This is because scientific investigation can only gather evidence for or against a particular phenomenon. Even theories such as gravity and natural selection have not been proven—they are simply supported by an almost endless amount of scientific evidence.

Last, let's use this information to solve the following problem:

A car manufacturer wants to produce a sports car that has the following average quarter mile run time:

Researchers decide to randomly sample two hundred cars off of the stock production line. They find that these cars' quarter mile run mean time is as follows:

Is it reasonable to say that the car performs to the manufacturer's specifications?

The researchers found that the mean time of two hundred cars was:

According to this sample statistic, it is reasonable to assume that the car can and will perform to the specifications of the manufacturer. This is because the calculated mean run time is only slightly above the desired average quarter mile run time. The time that was calculated in the experiment represents the test statistic, while the manufacturer's desired quarter mile run time represents the population parameter. The statistic represents a single experiment used to test a hypothesis that stated: the new car has an average quarter mile run time of

This hypothesis was tested by a single experiment that measured the quarter mile run time of two hundred stock cars. Replication of this experiment would gather means of multiple samples. If the researchers were able to gather a census of the quarter mile run time of the population, then they could create a standard curve:

A standard distribution represents a distribution of the means; furthermore, its mean (i.e. the mean of the means) is hypothetically equal to the population parameter mean.

It is costly and unlikely that a researcher will be able to take a census of all the instances of a particular phenomenon. As a result, many researchers will take a limited sample and fit a standard curve to it in order to gather a representative estimate of a populations mean—in this case quarter mile run times. In this example, researchers could gather the mean of one hundred experiments and create the following graph:

We can see in this graph that the sample statistic——hovers around the center—mean of the means—of the graph; therefore, it is reasonable to assume that this statistic supports the hypothesis that stated that the car could meet designer specifications.