91Ó°ÊÓ

Cold cabin? The Fathom screen shot below shows the results of taking $500$SRSs of $10$temperature readings from a population distribution that’s $N(50,3)$and recording the sample variance $2x$each time.

(a) Describe the approximate sampling distribution.

(b) Suppose that the variance from an actual sample is $s^{2}x=25$. What would you conclude about the thermostat manufacturer’s claim? Explain

Cold cabin? The Fathom screenshot below shows the results of taking $500$SRSs of $10$temperature readings from a population distribution that’s $N(50,3)$and recording the sample minimum each time.

(a) Describe the approximate sampling distribution.

(b) Suppose that the minimum of an actual sample is $40°F$. What would you conclude about the thermostat manufacturer’s claim? Explain.

The correlation between the age and height of children under the age of $12$is found to be$r=0.60$. Suppose we use the age$x$ of a child to predict the height y of the child. What can we conclude?
(a) The height is generally$60\%$ of a child’s weight.
(b) About$60\%$ of the time, age will accurately predict height.
(c) The fraction of the variation in heights explained by the least-squares regression line of $y$on $x$is$0.36$.
(d) The least-squares regression line of $y$on $x$has a slope of$0.6$.
(e) Thirty-six percent of the time, the least-squares regression line accurately predicts height.

About $75\%$of young adult Internet users (ages 18 to 29) watch online video. Suppose that a sample survey contacts an SRS of 1000 young adult Internet users and calculates the proportion $\hat{p}$in this sample who watch online video.

4. If the sample size were 9000 rather than 1000 , how would this change the sampling distribution of $\hat{p}$?

Run a mile During World War II, able-bodied male undergraduates at the University of Illinois participated in required physical training. Each student ran a timed mile. Their times followed the Normal distribution with mean $7.11$minutes and standard deviation $0.74$minute. An SRS of $100$of these students has mean time $\stackrel{¯}{x}=7.15$minutes. A second SRS of size$100$has mean $\stackrel{¯}{x}=6.97$minutes. After many SRSs, the values of the sample mean x follow the Normal distribution with mean $7.11$ minutes and standard deviation$0.074$minute.

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

An agronomist wants to test three different types of fertilizer (A, B, and C) on the yield of a new variety of wheat. The yield will be measured in bushels per acre. Six one-acre plots of land were randomly assigned to each of the three fertilizers. The treatment, experimental unit, and
response variable are, respectively,
(a) a specific fertilizer, bushels per acre, a plot of land.
(b) a plot of land, bushels per acre, a specific fertilizer.
(c) random assignment, a plot of land, bushels per acre.
(d) a specific fertilizer, a plot of land, bushels per acre.
(e) a specific fertilizer, the agronomist, bushels per acre

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 $pˆ$. How is it different from the population distribution?

According to the U.S. census, the proportion of adults in a certain county who owned their own home was $0.71$. An SRS of $100$adults in a certain section of the county found that$65$ owned their home. Which one
of the following represents the approximate probability of obtaining a sample of $100$adults in which fewer than $65$own their home, assuming that this section of the county has the same overall proportion of adults who own their home as does the entire county?

(a) $\left(\begin{array}{c}100\\ 65\end{array}\right)(0.71{)}^{65}(0.29{)}^{35}$

(b) $\left(\begin{array}{c}100\\ 65\end{array}\right)(0.29{)}^{65}(0.71{)}^{35}$
(c) $P\left(z<\frac{0.65-0.71}{\sqrt{\frac{(0.71)(0.29)}{100}}}\right)$

(d) $P\left(z<\frac{0.65-0.71}{\sqrt{\frac{(0.65)(0.35)}{100}}}\right)$
(e) $P\left(z<\frac{0.65-0.71}{\left.\frac{(0.71)(0.29)}{\sqrt{100}}\right)}\right.$

IRS audits The Internal Revenue Service plans to examine an SRS of individual federal income tax returns from each state. One variable of interest is the proportion of returns claiming itemized deductions. The total number of tax returns in each state varies from over 15 million in California to about $240,000$in Wyoming.

(a) Will the sampling variability of the sample proportion change from state to state if an SRS of $2000$tax returns is selected in each state? Explain your answer.

(b) Will the sampling variability of the sample proportion change from state to state if an SRS of $1\%$of all tax returns is selected in each state? Explain your answer

Which one of the following would be a correct interpretation if you have a z-score of $+2.0$on an exam?

(a) It means that you missed two questions on the exam.
(b) It means that you got twice as many questions correct as the average student.
(c) It means that your grade was two points higher than the mean grade on this exam.
(d) It means that your grade was in the upper $2\%$of all grades on this exam.
(e) It means that your grade is two standard deviations above the mean for this exam.