Q.10 - The Practice of Statistics for AP

Question

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. 

Step-by-Step Solution

Answer

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

b). Yes, there are $2$ outliers.

c). The claim appears to be true.

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 $\mu =64$ inches and standard deviation  $\sigma =2.5$ 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 $N(64,2.5)$.

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.