Q. 56 56. The SAT again High school st... [FREE SOLUTION]

91影视

The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset 91影视 Explanations$ Math

4th Edition 路 809 pages

ISBN: 9781319113339

Chapter 8: Q. 56 (page 518)

56. The SAT again High school students who take the SAT Math exam a second time generally score higher than on their first try. Past data suggest that the score increase has a standard deviation of about $50$ points.
How large a sample of high school students would be needed to estimate the mean change in SAT score to within $2$ points with $95 %$ confidence? Show your work.

Short Answer

Expert verified

The mean change in SAT score to within $2$ points with $95 %$ confidence is $n 鈮� 2401$ .

Step by step solution

Given information

Past data suggest that the score increase has a standard deviation of about $50$ points. To estimate the mean change in SAT score to
within $2$ points with $95 %$ confidence.

Explanation

Convert the confidence interval $95 %$ into decimal.
$\frac{95}{100} = 0.95$
Using table A, calculate the critical value:
$z_{伪 / 2} = z_{(1 鈭� 0.95) / 2} = 1.96$

Note that, Margin of error $E$ is $2,$ and, the standard deviation $蟽$ is $50$
Determine the sample size:

localid="1649838874210" $E 鈮� z_{伪 / 2} \frac{蟽}{\sqrt{n}}$

localid="1649838887825" $= 2 鈮� 1.96 \frac{50}{\sqrt{n}}$

localid="1649838893763" $= 2 脳 \sqrt{n} 鈮� 1.96 脳 50$

localid="1649838900904" $= \sqrt{n} 鈮� 49$

Remove the square from both sides:
$= (\sqrt{n})^{2} 鈮� 49^{2}$
$= n 鈮� 2401$

Hence, the sample size is $n 鈮� 2401$ .

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