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Q.10

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The Practice of Statistics for AP

Found in: Page 429

Book edition 4th

Author(s) David Moore,Daren Starnes,Dan Yates

Pages 809 pages

ISBN 9781319113339

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Short Answer

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 $μ = 64$ inches and standard deviation $σ = 2.5$ 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 $\bar{x}$ .

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

c). The claim appears to be true.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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" $μ = 64$ inches and standard deviation role="math" localid="1649581345730" $σ = 2.5$ 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 $μ = 64$ inches and standard deviation $σ = 2.5$ 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 $μ = 64$ inches and standard deviation $σ = 2.5$ 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.

Most popular questions for Math Textbooks

You want to take an SRS $50$ of the $816$ students who live in a dormitory on a college campus. You label the students $001$ to $816$ in alphabetical order. In the table of random digits, you read the entries

$95592$ $94007$ $69769$ $33547$ $72450$ $16632$ $81194$ $14873$

The first three students in your sample have labels

(a) $955, 929, 400$

(b) $400, 769, 769$

(c) $559, 294, 007$

(d) $929, 400, 769$

(e) $400, 769, 335$

Scooping beads A statistics teacher fills a large container with $1000$ white and $3000$ red beads and then mixes the beads thoroughly. She then has her students take repeated SRSs of $50$ beads from the container. After many SRSs, the values of the sample proportion pˆ of red beads are approximated well by a Normal distribution with mean of $0.75$ and standard deviation of $0.06$ .

(a) What is the population? Describe the population distribution. (b) Describe the sampling distribution of $p ˆ$ . How is it different from the population distribution?

In the language of government statistics, you are “in the labor force” if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) who are unemployed. Here are data from the Current Population Survey for the civilian population aged $25$ years and over in a recent year. The table entries are counted by thousands of people.

Unemployment (5.1) What is the probability that a randomly chosen person 25 years of age or older is in the labor force? Show your work.

The central limit theorem is important in statistics because it allows us to use the Normal distribution to make inferences concerning the population mean

(a) if the sample size is reasonably large (for any population).

(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).

(d) if the population is normally distributed and the population variance is known (for any sample size).

(e) if the population size is reasonably large (whether the population distribution is known or not,

A large company is interested in improving the efficiency of its customer service and decides to examine the length of the business phone calls made to clients by its sales staff. A cumulative relative frequency graph is shown below from data collected over the past year. According to the graph, the shortest $80 %$ of calls will take how long to complete?

(a) Less than $10$ minutes.
(b) At least $10$ minutes.(c) Exactly $10$ minutes.(d) At least $5.5$ minutes.(e) Less than $5.5$ minutes.

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