Q. 14 Tall girls? To see if the claim ... [FREE SOLUTION]

91影视

The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 837 pages

ISBN: 9781319113339

Chapter 7: Q. 14 (page 455)

Tall girls? To see if the claim made in Exercise $12$ is true at their high school, an Ap Statistics class chooses an SRS of twenty $16$ -year-old females at the school and measures their heights. In their sample, the mean height is $64.7$ inches. Does this provide convincing evidence that $16$ -year-old females at this school are taller than $64$ inches, on average?
a. What is the evidence that the average height of all $16$ -year-old females at this school is greater than $64$ inches, on average?
b. Provide two explanations for the evidence described in part (a).
We used technology to simulate choosing $250$ SRSs of size $n = 20$ from a population of three hundred $16$ -year-old females whose heights follow a Normal distribution with mean localid="1654113150676" $渭 = 64$ inches and standard deviation $渭 = 2.5$ inches. The dotplot shows $x =$ the sample mean height for each of the $250$ simulated samples.
c. There is one dot on the graph at $62.5$ . Explain what this value represents.
d. Would it be surprising to get a sample mean of $x = 64.7$ or larger in an SRS of size $20$ when $渭 = 64$ inches and $蟽 = 2.5$ inches? Justify your answer.
e. Based on your previous answers, is there convincing evidence that the average height of all $16$ -year-old females at this school is greater than $64$ inches? Explain your reasoning.

Short Answer

Expert verified

(a) The $64.7 inches$ is the sample mean, which is larger than data-custom-editor="chemistry" $64 inches$ .

(b) It's also feasible that the sample mean is bigger than $64$ inches because the population's average height is higher.

(c) One simple random sample of twenty females aged sixteen years old, with a mean height of data-custom-editor="chemistry" $62.5 inches$ .

(d) No, it is not surprising to get the given sample mean.

(e) No, there is no convincing evidence for the average height.

Step by step solution

Part (a) Step 1: Given information

We need to find the evidence for an average height of females.

Part (a) Step 2: Explanation

We know that

Sample mean is, $\overset{炉}{x} = 64.7$

Therefore, The sample mean of $64.7$ inches, which is greater than $64$ inches, is proof that the "average height of all $16$ -year-old females at this school is greater than $64$ inches, on average."

Part (b) Step 1: Given information

We need to find the explanations for the evidence described in part (a).

Part (b) Step 2: Explanation

Because the population mean height is $64$ inches and we got a sample with a sample mean of $64.7$ inches by chance, it's feasible that the sample mean is more than $64$ inches.

However, it's also feasible that the sample mean is bigger than $64$ inches because the population's average height is higher.

Part (c) Step 1: Given information

We need to find the representation of value $62.5$ .

Part (c) Step 2: Explanation

Each dot in the dotplot indicates a sample mean $x$ for a simple random sample (SRS) of twenty $16$ -year-old females, where the sample mean is the sample's average height.

The $62.5$ -inch dot indicates a simple random sample of twenty $16$ -year-old females with a sample mean height of $62.5$ inches.

Part (d) Step 1: Given information

We need to find out whether the value sample mean is surprising or not.

Part (d) Step 2: Explanation

In the above dotplot, there are $11$ dots above $64.7$ inches and several dots to the right of $64.7$ inches. When the population mean is $64$ inches, this means that a sample mean of at least $64.7$ inches is very likely to be obtained. As a result, a sample mean of $64.7$ or higher is not uncommon.