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A fast-food restaurant uses an automated filling machine to pour its soft drinks. The machine has different settings for small, medium, and large drink cups. According to the machine鈥檚 manufacturer, when the large setting is chosen, the amount of liquid dispensed by the machine follows a Normal distribution with mean $27$ounces and standard deviation$0.8$ounces. When the medium setting is chosen, the amount of liquid dispensed follows a Normal distribution with mean $17$ounces and standard deviation $0.5$ounces. To test the manufacturer鈥檚 claim, the restaurant manager measures the amount of liquid in a random sample of $25$cups filled with the medium setting and a separate random sample of $20$cups filled with the large setting. Let ${\stackrel{炉}{x}}_1-{\stackrel{炉}{x}}_2$be the difference in the sample mean amount of liquid under the two settings (large 鈥� medium). Find the mean and standard deviation of the sampling distribution.

A surprising number of young adults (ages $19$to $25$) still live in their parents鈥� homes. A random sample by the National Institutes of Health included $2253$men and $2629$women in this age group. The survey found that $986$of the men and $923$of the women lived with their parents.

(a) Construct and interpret a $99\%$confidence interval for the difference in population proportions (men minus women).

(b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Pat wants to compare the cost of one- and two-bedroom apartments in the area of her college campus. She collects data for a random sample of 10 advertisements of each type. The table below shows the rents (in dollars per month) for the selected apartments.

Pat wonders if two-bedroom apartments rent for significantly more, on average than one-bedroom apartments. She decides to perform a test of $H_0:{渭}_1={渭}_2$ versus $H_a:{渭}_1<{渭}_2$, where ${渭}_1$ and ${渭}_2$ are the true mean rents for all one-bedroom and two-bedroom aparaments, respectively, near the campus.

(a) Name the appropriate test and show that the conditions for carrying out this test are met.

(b) The appropriate test from part (a) yields a P-value of $0.058$. Interpret this P-value in context.

(c) What conclusion should Pat draw at the$伪=0.05$ significance level? Explain.

A fast-food restaurant uses an automated filling machine to pour its soft drinks. The machine has different settings for small, medium, and large drink cups. According to the machine鈥檚 manufacturer, when the large setting is chosen, the amount of liquid dispensed by the machine follows a Normal distribution with mean $27$ounces and standard deviation $0.8$ounces. When the medium setting is chosen, the amount of liquid dispensed follows a Normal distribution with mean $17$ounces and standard deviation $0.5$ounces. To test the manufacturer鈥檚 claim, the restaurant manager measures the amount of liquid in a random sample of $25$cups filled with the medium setting and a separate random sample of $20$ cups filled with the large setting. Let ${\overline{x}}_1-{\stackrel{炉}{x}}_2$be the difference in the sample mean amount of liquid under the two settings (large 鈥� medium). Find the probability that${\stackrel{炉}{x}}_1-{\stackrel{炉}{x}}_2$ is more than $12$ ounces. Show your work.

Which of the following is not a property of a binomial setting?

(a) Outcomes of different trials are independent.

(b) The chance process consists of a fixed number of trials,$n$

(c) The probability of success is the same for each trial.

(d) If we use a sample size of $30$, the binomial distribution will be approximately Normal.

(e) Each trial can result in either a success or a failure.

The elderly fear crime more than younger people, even though they are less likely to be victims of crime. One study recruited separate random samples of $56$black women and $63$black men over the age of $65$from Atlantic City, New Jersey. Of the women, $27$said they 鈥渇elt vulnerable鈥� to crime; $46$of the men said this.

(a) Construct and interpret a $90\%$confidence interval for the difference in population proportions (men minus women).

(b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Mrs. Woods and Mrs. Bryan are avid vegetable gardeners. They use different fertilizers, and each claims that hers is the best fertilizer to use when growing tomatoes. Both agree to do a study using the weight of their tomatoes as the response variable. They had each planted the same varieties of tomatoes on the same day and fertilized the plants on the same schedule throughout the growing season. At harvest time, they each randomly select $15$tomatoes from their respective gardens and weigh them. After performing a two-sample t test on the difference in mean weights of tomatoes, they get t $5.24$and P $0.0008.$ Can the gardener with the larger mean claim that her fertilizer caused her tomatoes to be heavier?

(a) No, because the soil conditions in the two gardens is a potential confounding variable.

(b) No, because there was no replication.

(c) Yes, because a different fertilizer was used on each garden.

(d) Yes, because random samples were taken from each garden.

(e) Yes, because the P-value is so low

Mrs. Woods and Mrs. Bryan are avid vegetable gardeners. They use different fertilizers, and each claims that hers is the best fertilizer to use when growing tomatoes. Both agree to do a study using the weight of their tomatoes as the response variable. They had each planted the same varieties of tomatoes on the same day and fertilized the plants on the same schedule throughout the growing season. At harvest time, they each randomly select $15$tomatoes from their respective gardens and weigh them. After performing a two-sample t-test on the difference in mean weights of tomatoes, they get $t=5.24$and $P=0.0008$. Can the gardener with the larger mean claim that her fertilizer caused her tomatoes to be heavier?

(a) No, because the soil conditions in the two gardens is a potential confounding variable.

(b) No, because there was no replication.

(c) Yes, because a different fertilizer was used on each garden.

(d) Yes, because random samples were taken from each garden.

As part of the Pew Internet and American Life Project, researchers surveyed a random sample of $800$teens and a separate random sample of $400$young adults. The teens, $79\%$said that they own an iPod or MP3 player. For the young adults, this figure was$67\%$. Is there a significant difference between the population proportions? State appropriate hypotheses for a significance test to answer this question. Define any parameters you use

A study by the National Athletic Trainers Association surveyed random samples of $1679$high school freshmen and $1366$high school seniors in Illinois. Results showed that $34$of the freshmen and $24$ of the seniors had used anabolic steroids. Steroids, which are dangerous, are sometimes used to improve athletic performance. Is there a significant difference between the population proportions? State appropriate hypotheses for a significance test to answer this question. Define any parameters you use.