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Q.10

Expert-verifiedFound in: Page 429

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 $N(64,2.5)$? To find out, we used Fathom software to simulate choosing $250$ SRSs of size $n=20$ students from a population that is $N(64,2.5)$. 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.

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.

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 $20$.

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.

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 $65.7$.

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.

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

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 $N(64,2.5)$.

Then it appears that the claim is true.

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