Your teacher brings two large bags of colored goldfish crackers to class. Bag 1 has 25% red crackers and Bag 2 has 35% red crackers. Using a paper cup, your teacher takes an SRS of 50 crackers from Bag 1 and a separate SRS of 40 crackers from Bag 2. Let $\hat{p}_{1}-\hat{p}_{2}$ be the difference in the sample proportions of red crackers. What is the shape of the sampling distribution of $\hat{p}_{1}-\hat{p}_{2} ?$ Why?

An SRS of 400 American adults is asked, "What do you think is the most serious problem facing our schools?" Suppose that in fact 40% of all adults would answer "violence" if asked this question. The proportion

$$\^{p}$$

of the sample who answer "violence" will vary in repeated sampling. In fact, we can assign probabilities to values of

$$\^{p}$$

using the Normal density curve with mean 0.4 and standard deviation 0.024. Use this density curve to find the probabilities of the event: (a) At least 45% of the sample believes that violence is the schools' most serious problem.

Americans have become increasingly concerned about the rising cost of Medicare. In $1990$, the average annual Medicare spending per enrollee was $\$3267$; in $2003$, the average annual Medicare spending per enrollee was $\$6883$ ($Money$, Fall $2003$). Suppose you hired a consulting firm to take a sample of fifty $2003$ Medicare enrollees to further investigate the nature of expenditures. Assume the population standard deviation for $2003$ was $\$2000.$

Show the sampling distribution of the mean amount of Medicare spending for a sample of fifty $2003$ enrollees.

We assume that we are obtaining simple random samples from infinite populations when obtaining sampling distributions. If the size of the population is finite, we technically need a finite population correction factor. However, if the sample size is small relative to the size of the population, this factor can be ignored. Explain what an "infinite population" is. What is the finite population correction factor? How small must the sample size be relative to the size of the population so that we can ignore the factor? Finally, explain why the factor can be ignored for such samples.

USA Today reported that about 47% of the general consumer population in the United States is loyal to the automobile manufacturer of their choice. Suppose Chevrolet did a study of a random sample of 1006 Chevrolet owners and found that 490 said they would buy another Chevrolet. Does this indicate that the population proportion of consumers loyal to Chevrolet is more than 47%? Use $\alpha=0.01$. What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic.

“In general, are you satisfied or dissatisfied with the way things are going in the United States at this time?” For this question of interest, the Gallup organization reported that for results based on this sample of 1039 adults, you can say with 95% confidence that the margin of error is

$$\pm 4$$

percentage points. The distribution of proportions of those who indicate they are satisfied for all possible samples of size n from the population is called the sampling distribution of the population for that statistic. Show that the simulation performed with n = 10 does not meet the normal condition and that the simulation performed with n = 100 does meet the normal condition.

An SRS of 400 American adults is asked, "What do you think is the most serious problem facing our schools?" Suppose that in fact 40% of all adults would answer "violence" if asked this question. The proportion

$$\^{p}$$

of the sample who answer "violence" will vary in repeated sampling. In fact, we can assign probabilities to values of

$$\^{p}$$

using the Normal density curve with mean 0.4 and standard deviation 0.024. Use this density curve to find the probabilities of the event: (b) Less than 35% of the sample believes that violence is the most serious problem.

The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $50 and the standard deviation is$20. The distribution is not Normal: many households pay a low rate as part of a bundle with phone or television service, but some pay much more for Internet only or for faster connections. A sample survey asks an SRS of 50 households with Internet access how much they pay. Let $\bar{x}$ be the mean amount paid. What are the mean and standard deviation of the sampling distribution of $\bar{x}?$

Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted 1019 adults. A newspaper article about the poll said, "Results have a margin of sampling error of plus or minus 3 percentage points."

The news article said that 65% of men, but only 43% of women, think that efforts to promote equality have gone far enough. Explain why we do not have enough information to give confidence intervals for men and women separately.

Assume that the populations are normally distributed and that independent sampling occurred.

$$\begin{matrix} \text{ } & \text{Sample 1} & \text{Sample 2}\\ \text{n} & \text{13} & \text{8}\\ \text{}{\bar{x}} & \text{96.6} & \text{98.3}\\ \text{s} & \text{3.2} & \text{2.5}\\ \end{matrix}$$

Test the hypothesis that $\mu_{1}<\mu_{2}$ at the $\alpha$=0.05 level of significance for the given sample data.

(a) Five cards are randomly chosen from a standard deck, one at a time with replacement. Let $X, Y,Z$ be the numbers of chosen queens, kings, and other cards. Find the joint PMF of $X, Y,Z$. (b) Find the joint PMF of $X$ and $Y$ . Hint: In summing the joint PMF of $X, Y,Z$ over the possible values of $Z$, note that most terms are 0 because of the constraint that the number of chosen cards is five. (c) Now assume instead that the sampling is without replacement (all 5-card hands are equally likely). Find the joint PMF of $X, Y,Z$. Hint: Use the naive definition of probability.

In this exercise you will use TI-83 simulate sampling from the following uniform distribution:

You will then perform a goodness of fit test to see if a randomly generated sample distribution comes from a population that is different from this distribution.

Use the randint function. to randomly generate 200 digits from 0 to 9 , and store these values in list $L_4$.

Select the best answer. An SRS of size 100 is taken from Population A with proportion 0.8 of successes. An independent SRS of size 400 is taken from Population B with proportion 0.5 of successes. The sampling distribution for the difference (Population A - Population B) in sample proportions has what mean and standard deviation? (a) mean = 0.3; standard deviation = 1.3 (b) mean = 0.3; standard deviation = 0.40 (c) mean = 0.3; standard deviation = 0.047 (d) mean = 0.3; standard deviation = 0.0022 (e) mean = 0.3; standard deviation = 0.0002

Suppose you fit the first-order multiple-regression model

$$y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \varepsilon$$

to n = 25 data points and obtain the prediction equation

$$\hat { y } = 6.4 + 3.1 x _ { 1 } + .92 x _ { 2 }$$

The estimated standard deviations of the sampling distributions of

$$\hat { \beta } _ { 1 } \text { and } \hat { \beta } _ { 2 } \text { are } 2.3 \text { and. } 27$$

respectively. Test

$$H _ { 0 } : \beta _ { 1 } = 0 \text { against } H _ { a } : \beta _ { 1 } > 0 .$$

Use

$$\alpha = .05$$