Adults aged 16 or older were assessed in three types of literacy: prose, document, and quantitative. The scores in document literacy were the same for 19- to 24-year-olds and for 40- to 49-year-olds. A random sample of scores from a later year showed the following statistics.

Age group	Mean score	Population standard deviation	Sample size
19-24	280	56.2	40
40-49	315	52.1	35
Make a $95 \%$ confidence interval for the true difference in mean scores for these two groups. What does your interval say about the claim that there is no difference in mean scores?

The limit

$$\lim _{n \rightarrow \infty} \frac{n !}{\sqrt{2 \pi} n^{n+1 / 2} e^{-n}}=1$$

says that the relative error in the approximation

$$n ! \approx \sqrt{2 \pi} n^{n+1 / 2} e^{-n}$$

approaches zero as $n$ increases . That is $n!$ grows at a rate comparable to $\sqrt{2} \pi n^{n+1 / 2} e^{-n}$. This result, known as Stirling's Formula is often very useful in applied mathematics and statistics. Use the identity $\ln (n !)=\sum_{j=1}^{n} \ln j$ and the increasing nature of ln to show that if $n \geq 1$

$$\int_{0}^{n} \ln x d x<\ln (n !)<\int_{1}^{n+1} \ln x d x$$

and hence that

$$n \ln n-n<\ln (n !)<(n+1) \ln (n+1)-n$$

You have already calculated the following sample statistics: sample mean for girls = 8.0 hours, sample MAD for girls = 4.0 hours, Now calculate the mean and MAD for the data in the sample of boys. (You can use the table below to help organize your work for computing the MAD.), sample mean for boys =, sample MAD for boys =

$$\begin{matrix} \text{Data Value} & \text{Distance from the Mean}\\ \text{5}\\ \text{3}\\ \text{1}\\ \text{1}\\ \text{0}\\ \text{13}\\ \text{0}\\ \text{3}\\ \text{10}\\ \text{0}\\ \end{matrix}$$

Use a graphing calculator or other technology to do the problem.

The amount of U.S. expenditures on national defense (in billions of dollars) can by approximated by the function

$$f(x)=\frac{1084}{1+1.94 e^{-.171 x}},$$

where x = 0 corresponds to the year 2000.

(a) Estimate the expenditures in the years 2005 and 2010.

(b) Assume the model remains accurate and graph the function from the year 2000 to the year 2020.

(c) Use the graph to determine the year in which expenditures reach $850 billion.

Find the indicated probabilities using the Poisson distribution. If convenient, use a Poisson probability table or technology tool to find the probabilities.

The mean number of bankruptcies filed per minute in the United States in a recent year was about three. Find the probability that a) exactly five businesses will file bankruptcy in any given minute, b) at least five businesses will file bankruptcy in any given minute, and c) more than five businesses will file bankruptcy in any given minute.

Match the histograms on the right to the summary statistics given.

A)

B)

C)

D)

Over the past fifty years the average American's leisure time has increased by between 4 and 8 hours a week. Some economists think that this increase is primarily driven by a rise in wage rates. a. Use the income and substitution effects to describe the labor supply for the average American. Which effect dominates? b. In addition to increasing wages, a study by the Bureau of Labor Statistics finds labor force participation for women is projected to steadily increase through 2024. For the average woman who has entered the labor force, which c. Draw typical individual labor supply curves that illustrate your answers to part $a$ and part b above.

Independent random samples of bachelor’s and master’s degree holders in statistics, whose initial job was with a major actuarial firm and who subsequently moved to an insurance company, were questioned. For a sample of 44 bachelor’s degree holders, the mean number of months before the first job change was 35.02 and the sample standard deviation was 18.20. For a sample of 68 master’s degree holders, the mean number of months before the first job change was 36.34 and the sample standard deviation was 18.94. Test, at the 10% level against a two-sided alternative, the null hypothesis that the population mean numbers of months before the first job change are the same for the two groups.

Mutually exclusive versus independent. The two-way table summarizes data on the gender and eye color of students in a college statistics class. Imagine choosing a student from the class at random. Define event A: student is male, and event B: student has blue eyes.

$$\text{Gender}\\ \text{Eye color}\begin{array}{l|c|c|c} & \text { Male } & \text { Female } & \text { Total } \\ \hline \text { Blue } & & & 10 \\ \hline \text { Brown } & & & 40 \\ \hline \text { Total } & 20 & 30 & 50 \end{array}$$

Copy and complete the two-way table so that events A and B are independent.

A new production technology for making vitamins is invented by a college professor who decides not to patent it. Thus, it is available for anybody to copy and put into use. The TC per bottle for production up to $100,000$ bottles per day is given in the following table.

$$\begin{array}{rr} \text { Output } & \text { TC } \\ \hline 25,000 & \$ 50,000 \\ 50,000 & 70,000 \\ 75,000 & 75,000 \\ 100,000 & 80,000 \end{array}$$

Do both production technologies show constant returns to scale?

Insurance companies and other organizations use actuarial tables to estimate the remaining lifespans of their customers. The data file gives life expectancy and estimated additional years of life for black 111ales in the United States, according to a 2016 National Vital Statistics Report.

a) Fit a regression model to predict Life expectancy from Age. Does it look like a good fit? Now plot the residuals.

b) Find a re-expression to create a better model. Predict the life expectancy of an 18-year-old black man.

c) Are you satisfied that your model has accounted for the relationship between Life expectancy and Age? Explain.

An essential tool in the monitoring of the quality of a manufactured product is acceptance sampling . An acceptance sampling plan involves knowing the distribution of the life length of the item produced and determining how many items to inspect from the manufacturing process. The Journal of Applied Statistics (Apr. 2010) demonstrated the use of the exponential distribution as a model for the life length x of an item (e.g., a bullet). The article also discussed the importance of using the median of the lifetime distribution as a measure of product quality, since half of the items in a manufactured lot will have life lengths exceeding the median. For an exponential distribution with mean

$$\theta$$

give an expression for the median of the distribution. [Hint: Your answer will be a function of

$$\theta$$

.]

Use a graphing calculator or other technology to complete the exercise. You may need to find a larger viewing window for some problem.

Similar to the previous exercise, the amount of outstanding debt for home mortgages (in trillions of dollars) can be approximated by $-.15 x^2+2.53 x+.66$ where x = 4 corresponds to the year 2004. Assuming the model holds up to and including 2014, find the years in which the amount of outstanding home mortgage debt was higher than $9 trillion.

Married men tend to earn more than single men. An investigation of the relationship between marital status and income collected data on all 8235 men employed as managers or professionals by a large manufacturing firm in 1976. Suppose (this is risky) we regard these men as a random sample from the population of all men employed in managerial or professional positions in large companies. Here are descriptive statistics for the salaries of these men:

What are the degrees of freedom of the ANOVA F test?