We need to calculate Natalie's z-scores for both her Spanish and math exams. Calculating z-scores is as follows:

Her z-score for the Spanish exam is , which equals 1.25, while her z-score for the math exam is , which equals 1.50. Since both z-scores are positive, Natalie did above average on both tests, but since her z-score for the math exam is higher than her z-score for the Spanish exam, Natalie did better on her math exam when compared to the rest of her peers taking the exam.

To find the z-score, we follow the formula

 where  is the given value,  is the mean, and  is the standard deviation.

For this problem we see that

    and 

Substituting for these values we see

 is the average life of the bread maker, in this problem 9 years.

 is the standard deviation of the break makers life. The standard deviation is the square root of the variance. 

X is the test statistic we are looking to find. In this problem it is the probability that the bread maker lasts less than 13 years.

If you were to find the value associated with a z-score of 2 on the z-table, you would see that

A t test is used to compare the means of different groups.  A t test score describes the likelihood that the difference in means between two groups is due to chance. The null hypothesis assumes the two sets are equal however, one can reject the null hypothesis with a p value within a particular confidence level.

Use the fact that the sum of  squared standard and independent normal variables follows a chi-squared distribution with degrees of freedom.

Because we are only given the sample standard deviation we will use the t-distribution to calculate the confidence interval.

Appropriate Formula: 

Now we must identify our variables:

We must find the appropriate t-value based on the given

t-value at 90% confidence:

Look up t-value for 0.05, 55 , so t-value= ~ 1.6735

90% CI becomes:

First you must calculate the sample mean and sample standard deviation of the sample.

Because we do not know the population standard deviation we will use the t-distribution to calculate the confidence intervals. We must use standard error in this formula because we are working with the standard deviation of the sampling distribution.

Formula: 

To find the appropriate t-value for 95% confidence interval: 

Look up in t-table and the corresponding t-value = 2.093.

Thus the 95% confidence interval is: 

For there to be a statistically significant difference in the training programs, the 95% confidence interval cannot include zero.  includes zero, so we can't say that one program is significantly better than the other.

The subset symbol should be read as "is a subset of" So it's the first letter is a subset of the second letter. To be a subset, all of its elements must be contained in the other set.

The only one of these relationships that is true (where the entire set on the left is in the right set) is

The elements of B, 4, 5, 6, and 8 all appear in set D.

Step 1: A subset of a set must have elements in Set A. If any number in the subset is not in the original set, then that subset is not a subset of that set.

{} is not a subset because  is not in Set A.