Question

The distribution of scores on the mathematics part of the SAT exam in a recent year was approximately Normal with mean 515 and standard deviation 114 . Imagine choosing many SRSs of 100 students who took the exam and averaging their SAT Math scores. Which of the following are the mean and standard deviation of the sampling distribution of ${\mathrm{x}}^{-}$?$\overline{x}$?

a. Mean $=515,\mathrm{SD}=114$

b. Mean $=515,\mathrm{SD}=114/{100}^{\mathrm{SD}}=114/\sqrt{100}$

c. Mean role="math" localid="1654342976765" $=515/100,\mathrm{SD}=114/100$

d. Mean role="math" localid="1654343050312" $=515/100,\mathrm{SD}=114/100\mathrm{SD}=114/\sqrt{10}$

e. Cannot be determined without knowing the 100 scores.

Short answer

The value of Mean $=515$and SD$=\frac{114}{\sqrt{100}}$

The correct option is (b)

Step-by-step solution

Given information 

Given,

$$\begin{gathered} \mu =515 \\ \sigma =114 \\ n=100 \end{gathered}$$

Explanation for correct option

Calculation:

$$\begin{gathered} {\mu }_{\overline{x}}=\mu =515 \\ {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}=\frac{114}{\sqrt{100}} \end{gathered}$$

Hence, the correct option is (b)

Explanation for incorrect option

(a). mean$=515,\mathrm{SD}=114$are not the mean and standard deviation of the sampling distribution of role="math" localid="1654343011275" ${\mathrm{x}}^{-}$and role="math" localid="1654343023938" $\overline{x}$

(c). Mean$=515/100,\mathrm{SD}=114/100$are not the mean and standard deviation of the sampling distribution of role="math" localid="1654343077524" ${\mathrm{x}}^{-}$and role="math" localid="1654343088243" $\overline{x}$

(d). Mean$=515/100,\mathrm{SD}=114/100\mathrm{SD}=114/\sqrt{10}$are not the mean and standard deviation of the sampling distribution of ${\mathrm{x}}^{-}$and $\overline{x}$

(e). Cannot be determined without knowing the 100 scores is incorrect statement