Chapter 7: Q. 73 (page 481)

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 $x^{-}$ ? $\bar{x}$ ?
a. Mean $= 515, SD = 114$
b. Mean $= 515, SD = 114 / 100^{SD} = 114 / \sqrt{100}$
c. Mean role="math" localid="1654342976765" $= 515 / 100, SD = 114 / 100$
d. Mean role="math" localid="1654343050312" $= 515 / 100, SD = 114 / 100 SD = 114 / \sqrt{10}$
e. Cannot be determined without knowing the 100 scores.

Short Answer

Expert verified

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

The correct option is (b)

Step by step solution

Given information

Given,

μ = 515 σ = 114 n = 100

Explanation for correct option

Calculation:

$μ_{\bar{x}} = μ = 515 σ_{\bar{x}} = \frac{σ}{\sqrt{n}} = \frac{114}{\sqrt{100}}$

Hence, the correct option is (b)

Explanation for incorrect option

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

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

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