The data set represents the scores of 12 randomly selected stu...

Question

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

c. Would it be unusual for the population mean to be under 575? Explain.

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c. Would it be unusual for the population mean to be under 575? Explain.

Accepted Answer

Hello, everyone, let's take a look at this question together. Suppose the following are the test scores for 8 students, 540, 555, 560, 570, 565, 550, 575, and 560. Assuming that the population of test scores is normally distributed and that the population standard deviation is 72, would it be unusual for the population mean to be below 550? Explain your answer. So, in order to solve this question, we have to recall what we have learned about the population mean to determine if it would be unusual for the population of test scores that is normally distributed with a population standard deviation of 72 to be below 550. And so the first step in solving this problem is to find the sample mean. Which can be done so by adding all of the scores together and dividing by 8. So, our sample mean is calculated as all of the eight test scores, divided by 8, which, when simplified, is 4,475 divided by 8, which is equal to 559.375, so 559. 375 is our sample mean, and then we can identify the population standard deviation and the sample size, which we are given the population standard deviation is equal to 72 and the sample size N is equal to 8. And using these values, we can calculate the standard error of the mean. Using the formula for the standard error, which is equal to the standard deviation divided by the square root of our sample size, which substituting the values into the formula, we have 72 divided by the square root of 8, which is equal to 18 multiplied by the square root of 2. And now we can calculate the Z score for a population mean of 550. Using the formula for the Z score, which is equal to the sample mean subtracted by the population mean divided by the standard error of the mean, which substituting the values into the formula, we have 559.375, subtracted by 550 divided by 18, multiplied by the square root of 2, which is approximately 0.3682. So our Z score for a population mean of 550 is approximately 0.3682, which using the Z score, we can then interpret the value where we know that Z scores less than -2 or greater than 2 are considered unusual, and since 0.3682. Is between 2 and 2, the population mean to be below 550 is not unusual. Therefore, the population mean to be below 550 is in fact not unusual because the Z score is greater than -2 and less than 2, which makes answer choice A the correct answer. And all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Normal Distribution

Population Mean

Standard Deviation

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