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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 =64inches and standard deviation =2.5inches. To see if this distribution applies at their high school, an AP Statistics class takes an SRS of 20of the 30016-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 250SRSs of size n=20students 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 20girls in the class鈥檚 actual sample is x=64.7. What would you conclude about the population mean height Mfor the 16-year-old females at the school? Explain.

Short Answer

Expert verified

a). No, this is not a sampling distribution of x.

b). Yes, there are2outliers.

c). The claim appears to be true.

Step by step solution

01

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" =64inches and standard deviationrole="math" localid="1649581345730" =2.5 inches.

02

Part (a) Step 2: Explanation 

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

03

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 =64 inches and standard deviation =2.5 inches.

04

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 2outliers (one on each side of the dot-plot): 62.5 and 65.7.

05

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 =64 inches and standard deviation =2.5 inches.

06

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.

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Most popular questions from this chapter

More on insurance An insurance company knows that in the entire population of homeowners, the mean annual loss from 铿乺e is =250and the standard deviation of the loss is =300. The distribution of losses is strongly right-skewed: many policies have 0loss, but a few have large losses. If the company sells 10,000 policies, can it safely base its rates on the assumption that its average loss will be no greater than 275? Follow the four-step process

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(c) Suppose that 45of the100students in the actual sample say that they did all their homework last week. What would you conclude about the newspaper article鈥檚 claim? Explain.

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(b) if the population is normally distributed and the sample size is reasonably large.

(c) if the population is normally distributed (for any sample size).

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