This question relates to the  rule of normal distribution. We know that  of the data are within  standard deviations from the mean.

In this case this means that  of the students are between

 and  

 and 

 and 

Therefore we have  of the students that are outside of this range. Since the normal distribution is symmetric, the proportion of students below  is the same as the proportion of students above .

Thus the right answer is  or .

Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations of the mean.

In this case, 95% of the students' scores were between:

75-(2 x 4) and 75+(2 x 4)

or between a 67 and a 83, with equal amounts of the leftover 5% of scores above and below those scores. This would mean that 2.5% of the students scored below a 67% on the test.

Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations (in either direction) of the mean.

In this case, 99.7% of the students' scores were between 3 standard deviation above the mean and 3 standard deviations below the mean:

78-(2 x 3) and 78+(2 x 3)

or between a 72 and an 84. 

The graph of a normally-distributed data set is symmetrical.

The graph of a normally-distributed data set has a single, central peak at the mean of the data set that it describes.

The graph of a normally-distributed data set will vary based only upon the mean and the standard deviation of the set that it describes.

The graph of a normally-distributed data set with a higher standard deviation will be wider than the graph of a normally-distributed data set with a lower standard deviation.

The question asks us to find the statement that is not true; however, all statements are true so the correct response is "All of these are true."

If the mean of a normally distributed set of scores is  and the standard deviation is , then the -score corresponding to a test score of  is 

From a -score table, in a normal distribution, 

We want the percent of students whose test score is 85 or better, so we want . This is

or about 5.7 % The correct choice is 6%.

Let X represent the salaries of employees at XYZ Corporation.

We want to determine the probability that X is between 69,000 and 78,000:

To approximate this probability, we convert 69,000 and 78,000 to standardized values (z-scores).

We then want to determine the probability that z is between 1.2 and 2.4

The proportion of employees who earn between 69,000 and 78,000 is 0.107.

The z-score is a measure of an actual score's distance from the mean in terms of the standard deviation. The formula is:

Where  are the mean and standard deviation, respectively. is the actual score.

If we plug in the values we have from the original problem we have 

which is approximately .

The z-score can be expressed as

where 

Therefore the z-score is:

Use the formula for z-score:

Where  is her test score,  is the mean, and  is the standard deviation.

The formula for a z-score is

where   = mean and  = standard deviation and =your test grade.

Plugging in your z-score, mean, and standard deviation that was originally given in the question we get the following.

Now to find the grade you got on the test we will solve for .