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Expert-verified Found in: Page 429 ### The Practice of Statistics for AP

Book edition 4th
Author(s) David Moore,Daren Starnes,Dan Yates
Pages 809 pages
ISBN 9781319113339 # Tall girls According to the National Center for Health Statistics, the distribution of heights for 16-year-old females is modeled well by a Normal density curve with mean $\mu =64$ inches and standard deviation $\sigma =2.5$ inches. To see if this distribution applies at their high school, an AP Statistics class takes an SRS of $20$ of the $300$ $16$-year-old females at the school and measures their heights. What values of the sample mean x would be consistent with the population distribution being $N\left(64,2.5\right)$? To find out, we used Fathom software to simulate choosing $250$ SRSs of size $n=20$ students from a population that is $N\left(64,2.5\right)$. The figure below is a dotplot of the sample mean height x of the students in the sample. (a) Is this the sampling distribution of $x$? Justify your answer. (b) Describe the distribution. Are there any obvious outliers? (c) Suppose that the average height of the $20$ girls in the class’s actual sample is $x=64.7$. What would you conclude about the population mean height $M$ for the $16$-year-old females at the school? Explain.

a). No, this is not a sampling distribution of $\overline{x}$.

b). Yes, there are $2$ outliers.

c). The claim appears to be true.

See the step by step solution

## Part (a) Step 1: Given Information

According to the National Center for Health Statistics, the distribution of heights for $16$-year-old females is modeled well by a Normal density curve with mean role="math" localid="1649581331434" $\mu =64$ inches and standard deviation role="math" localid="1649581345730" $\sigma =2.5$ inches.

## Part (a) Step 2: Explanation

No, because the dotplot contains the results of $250$ simple random samples of size $20$, while the sampling distribution should contain the results of all possible samples of size $20$.

## Part (b) Step 1: Given Information

According to the National Center for Health Statistics, the distribution of heights for $16$-year-old females is modeled well by a Normal density curve with mean $\mu =64$ inches and standard deviation $\sigma =2.5$ inches.

## Part (b) Step 2: Explanation

Shape: Roughly unimodal and symmetric, because the highest peak is roughly in the middle of the histogram

Center: The highest peak in the histogram is at about $64.0$, thus the distribution is centered at $64.0$.

Spread: The data values appear to vary from $62.5$ to $65.7$.

Outliers are dots that are separated from the other dots in the dot-plot by a gap.

Then we note that there might be $2$ outliers (one on each side of the dot-plot): $62.5$ and $65.7$.

## Part (c) Step 1: Given Information

According to the National Center for Health Statistics, the distribution of heights for $16$-year-old females is modeled well by a Normal density curve with mean $\mu =64$ inches and standard deviation $\sigma =2.5$ inches.

## Part (c) Step 2: Explanation

Claim: The population distribution is $N\left(64,2.5\right)$.

In the dotplot we note that there are a lot of dots above $64.7$ and also a lot of dots to its right, this means that it is likely to obtain a sample mean of $64.7$ if the population distribution is $N\left(64,2.5\right)$.

Then it appears that the claim is true.

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