Question

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" $\mu =64$inches and standard deviation $\mu =2.5$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 $\mu =64$inches and $\sigma =2.5$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

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

(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" $62.5\mathrm{inches}$.

(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,$\overline{x}=64.7$

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.

Part (e) Step 1: Given information

We need to find the convincing evidence for the part (d).

Part (e) Step 2: Explanation

From part (d)

We know that

It is not surprising to get the sample mean greater than or equal to the given sample mean.

Which means that there is no persuasive evidence that all $16$-year-old females at this school are taller than $64$inches.

Therefore, there is no evidence.