Question

Question 15 of 20 (1 point Attempt 1 of 1 | View question in a popup 2h 30m Romaining 7.3 Section Exercise 24 (calc) SAT scores: Assume that in a given year the mean mathematics SAT score was 467, and the standard deviation was 105. A sample of 59 scores is chosen. Use the TI-84 Plus calculator. Part 1 of 5 (a) What is the probability that the sample mean score is less than 4557 Round the answer to at least four decimal places. The probability that the sample mean score is less than 455 is . Part 2 of 5 (b) What is the probability that the sample mean score is between 430 and 470? Round the answer to at least four decimal places. The probability that the sample mean score is between 430 and 470 is . Submit © 2019 McGraw-Hill Education. All Rights Reserved. Terms of X HH N

Answer 1

Solution

given : = 467, 0 = 105, n = 59

formula : Z=-

(a) Pli < 455) =?

I-4 455 – 467

P(Z < -0.88)

Refer standard normal table (Z - table) to find probability

一 0.1900

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(b) P(430< i < 470) =?

I >P 430 - 467 105 59 < -u 0 470 – 467 105 105

$\Rightarrow P\left (-2.71<Z<0.22 \right )$

$\Rightarrow P\left (Z<0.22 \right )-P(Z<-2.71)$

Refer standard normal table (Z - table) to find probability

$\Rightarrow 0.5869-0.0034$

$\Rightarrow \mathbf{{\color{Red} 0.5835}}$

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(c) 45th percentile P(Z <?) = 0.45

From Z-table, Lookup for Z-value corresponding to area 0.45 to the left of the normal curve.

$\Rightarrow P(Z<\mathbf{-0.126})=0.45$

$-0.126=\frac{\bar x-467}{\frac{105}{\sqrt{59}}}$

$\bar x=\mathbf{{\color{Red} 465.2776}}$

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${\color{Blue} \mathbf{(d)}}\;P(\bar x>499)=?$

$\Rightarrow P\left (\frac{\bar x-\mu}{\frac{\sigma }{\sqrt{n}}}>\frac{499-467}{\frac{105}{\sqrt{59}}} \right )$

> P(Z > 2.34)

$\Rightarrow 1-P\left (Z<2.34 \right )$

Refer standard normal table (Z - table) to find probability

$\Rightarrow 1-0.9904$

$\Rightarrow \mathbf{0.0096{\color{Red} }}$

It _{$\mathbf{{\color{Red} Would}}$} be unusual if the sample mean were greater than 499, since the probability is $\mathbf{{\color{Red} 0.0096}}$

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${\color{Blue} \mathbf{(e)}}\;P(X>499)=?$

$\mathbf{formula:}\;Z=\frac{X-\mu}{\sigma}$

$\Rightarrow P\left (\frac{X-\mu}{\sigma}>\frac{499-467}{105} \right )$

$\Rightarrow P\left (Z>0.30 \right )$

>1-P(Z <0.30)

Refer standard normal table (Z - table) to find probability

$\Rightarrow 1-0.6197$

$\Rightarrow \mathbf{0.3803}$

_{${\color{red} \mathbf{Yes}},$}because the probability that an individual gets a score greater than 499 is ${\color{red} \mathbf{0.3803}}$