Question

question 8. suppose a statistics professor reports a final exam average of 82 with a standard deviation of 4 points. which of these correctly explains where a score of 72 falls within the distribution? a. it is within 3 standard deviations of the mean. b. it is exactly 2 standard deviations from the mean. c. it is more than 3 standard deviations from the mean. d. it is exactly 3 standard deviations from the mean.

Explanation:

Step1: Calculate the difference from the mean

The mean ($\mu$) is 82, and the score is 72. The difference is $|72 - 82| = 10$.

Step2: Calculate how many standard deviations this is

The standard deviation ($\sigma$) is 4. To find the number of standard deviations, we calculate $\frac{10}{4} = 2.5$.

Step3: Analyze the result

2.5 is more than 2 and less than 3. So, 72 is more than 2 standard deviations from the mean but within 3 standard deviations (since 2.5 < 3). Now let's check the options:

• Option A: It is within 3 standard deviations of the mean. Since 2.5 < 3, this is correct.
• Option B: It is exactly 2 standard deviations from the mean. 2.5 ≠ 2, so incorrect.
• Option C: It is more than 3 standard deviations from the mean. 2.5 < 3, so incorrect.
• Option D: It is exactly 3 standard deviations from the mean. 2.5 ≠ 3, so incorrect.

Answer:

A. It is within 3 standard deviations of the mean.