Chapter 7: Sampling Distributions

Healthy Living From a large group of people who signed a card saying they intended to quit smoking, $1000$ people were selected at random. It turned out that $210(21\%)$ of these individuals had not smoked over the past $6$ months.

Sample medians List all $10$possible SRSs of size $\mathrm{n}=3$, calculate the median quiz score for each sample, and display the sampling distribution of the sample median on a dotplot.

Doing homework A school newspaper article claims that $60\%$of the students at a large high school completed their assigned homework last week. Assume that this claim is true for the $2000$students at the school.

a. Make a bar graph of the population distribution.

b. Imagine one possible SRS of size $100$from this population. Sketch a bar graph of the distribution of sample data.

Tall girls According to the National Center for Health Statistics, the distribution of height for $16$-year-old females is modeled well by a Normal density curve with mean $\mathrm{\mu }=64$inches and standard deviation $\mathrm{\sigma }=2.5$inches. Assume this claim is true for the three hundred $16$-year-old females at a large high school.

a. Make a graph of the population distribution.

b. Imagine one possible SRS of size $20$from this population. Sketch a dotplot of the distribution of sample data.

More homework Some skeptical Ap® Statistics students want to investigate the newspaper's claim in Exercise $11$, so they choose an SRS of $100$students from the school to interview. In their sample, $45$students completed their homework last week. Does this provide convincing evidence that less than $60\%$of all students at the school completed their assigned homework last week?

a. What is the evidence that less than $60\%$of all students completed their assigned homework last week?

b. Provide two explanations for the evidence described in part (a).

We used technology to simulate choosing $250$SRSs of size $n={100}^{n=100}$from a population of $2000$students where $60\%$completed their assigned homework last week. The dotplot shows $p^{\hat{p}}$the sample proportion of students who completed their assigned homework last week for each of the $250$simulated samples.

c. There is one dot on the graph at $0.73$. Explain what this value represents.

d. Would it be surprising to get a sample proportion of $p=0.{45}^{\hat{p}=0.45}$or smaller in an SRS of size $100$when $p=0.{60}^{p=0.60}$? Justify your answer.

e. Based on your previous answers, is there convincing evidence that less than $60\%$of all students at the school completed their assigned homework last week? Explain your reasoning.

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.

Suppose that the sample proportion of students who did all their assigned homework last week is $\hat{p}=\frac{57}{100}=0.57$. Would this sample proportion provide convincing evidence that less than $60\%$of all students at the school completed all their assigned homework last week? Explain your reasoning.

Even more tall girls, Refer to Exercises $12$and $14$. Suppose that the sample mean height of the twenty $16$-year-old females is $\overline{x}=65.8$inches. Would this sample mean provide convincing evidence that the average height of all $16$-year-old females at this school is greater than $64$inches? Explain your reasoning.

Cold cabin? The dotplot shows the results of taking $300$SRSs of $10$temperature readings from a Normal population with ${\mu }^{\mu }=50$and ${\sigma }^{\sigma }=3$and recording the sample standard deviation $s{x}^{s_x}$each time. Suppose that the standard deviation from an actual sample is $sx=5°F{.}^{sx=5°F}$. What would you conclude about the thermostat manufacturer's claim? Explain your reasoning.

Really cold cabin The dotplot shows the results of taking $300$SRSs of $10$temperature readings from a Normal population with $\mu =50$and $\sigma =3$and recording the sample minimum each time. Suppose that the minimum of an actual sample is $40°F$. What would you conclude about the thermostat manufacturer’s claim? Explain your reasoning.