Last year's records of auto accidents occurring on a given section of highway were classified according to whether the resulting damage was $1000 or more and to whether a physical injury resulted from the accident. The data follows:

Under$1000: Number of Accidents 32 and Number Involving injuries 10 $1000 or more: Number of Accidents 41 and Number Involving Injuries 23

a. Estimate the true proportion of accidents involving injuries when the damage was$1000 or more for similar sections of highway and find the margin of error. b. Estimate the true difference in the proportion of accidents involving injuries for accidents with damage under $1000 and those with damage of$1000 or more. Use a 95% confidence interval.

Find the appropriate rejection regions for the large-sample test statistic z in these cases: a. A left-tailed test at the 1% significance level. b. A two-tailed test with $\alpha$=.01. c. Suppose that the observed value of the test statistic was z=-2.41. For the rejection regions constructed in parts a and b, draw the appropriate conclusion for the tests. If appropriate, give a measure of the reliability of your conclusion.

Bethany has been working full time for one year and frequently finds herself short of cash. Her total income is $39,000, and her income after taxes is$2,500 per month. She has the following expenses. Assume there are four weeks in a month.

• Car payment, $400/month
• Auto insurance,$3,000/year
• Gas, $SO/week
• Groceries,$150/week
• Clothing, $50/week
• Entertainment,$25/week
• Utilities, (monthly average) $150/month
• Rent,$700/month
• Cell phone, $30/per month

1. Based on "pay yourself first," calculate the amount that Bethany should budget for savings.

Refer to the mentioned data set and find the sample proportion of M&Ms that are red. Use that result to test the claim of Mars, Inc., that 20% of its plain M&M candies are red.

Two systems $\mathrm { K }$ and $\mathrm { K } ^ { \prime }$ synchronize their clocks at $t =$ $t ^ { \prime } = 0$ when their origins are aligned as system $\mathrm { K } ^ { \prime }$ passes by system $\mathrm { K }$ along the $x$ axis at relative speed $0.8 \: \mathrm { c }$ . At time $t = 3 \: \mathrm { ns } ,$ Frank $($ in system $\mathrm { K } )$ shoots a proton gun having proton speeds of $0.98 \: c$ along his $x$ axis. The protons leave the gun at $x = 1 \: \mathrm { m }$ and arrive at a target $120 \: \mathrm { m }$ away. Determine the event coordinates $( x , t )$ of the gun firing and of the protons arriving as measured by observers in both systems $\mathrm { K }$ and $\mathrm { K } ^ { \prime }$ .

Does Mars, Incorporated use the same proportion of red candies in its plain and peanut varieties? A random sample of 56 plain M&M's contained 12 red candies, and another random sample of 32 peanut M&M's contained 8 red candies. a. Construct a 95% confidence interval for the difference in the proportions of red candies for the plain and peanut varieties. b. Based on the confidence interval in part a, can you conclude that there is a difference in the proportions of red candies for the plain and peanut varieties? Explain.

Independent random samples of $n_1=n_2=n$ observations are to be selected from each of two populations 1 and 2. If you wish to estimate the difference between the two population means correct to within .17, with probability equal to .90, how large should $n_1$ and $n_2$ be? Assume that you know $\sigma_1^2\approx \sigma_2^2\approx$27.8.

Even within a particular chain of hotels, lodging during the summer months can vary substantially depending on the type of room and the amenities offered. Suppose that we randomly select 50 billing statements from each of the computer databases of the Marriott, Westin, and the Doubletree hotel chains, and record the nightly room rates.

Marriott: Sample average $150 and sample standard deviation$17.2 Westin: Sample average $165 and sample standard deviation$22.5 Doubletree: Sample average $125 and sample standard deviation$12.8

a. Describe the sampled population(s). b. Find a point estimate for the average room rate for the Marriott hotel chain. Calculate the margin of error. c. Find a point estimate for the average room rate for the Westin hotel chain. Calculate the margin of error. d. Find a point estimate for the average room rate for the Doubletree hotel chain. Calculate the margin of error. e. Display the results of parts b, c, and d graphically. Use this display to compare the average room rates for the three hotel chains.

Independent random samples of $n_1$=800 and $n_2$=640 observations were selected from binomial populations 1 and 2, and $x_1$=337 and $x_2$=374 successes were observed. a. Find a 90% confidence interval for the difference $(p_1-p_2)$ in the two population proportions. Interpret the interval. b. What assumptions must you make for the confidence interval to be valid? Are these assumptions met?

A sample survey is designed to estimate the proportion of sports utility vehicles being driven in the state of California. A random sample of 500 registrations are selected from a Department of Motor Vehicles database, and 68 are classified as sports utility vehicles. a. Use a 95% confidence interval to estimate the proportion of sports utility vehicles in California. b. How can you estimate the proportion of sports utility vehicles in California with a higher degree of accuracy? (HINT: There are two answers.)

Suppose you wish to estimate a population mean based on a random sample of n observations, and prior experience suggest that $\sigma$=12.7. If you wish to estimate $\mu$ correct to within 1.6, with probability equal to .95, how many observations should be included in your sample?

Find the appropriate rejection regions for the large-sample test statistic z in these cases: a. A right-tailed test with $\alpha$=.01 b. A two-tailed test at the 5% significance level

In a study of the relationship between birth order and college success, an investigator found that 126 in a sample of 180 college graduates were firstborn or only children. In a sample of 100 nongraduates of comparable age and socioeconomic background, the number of firstborn or only children was 54. Estimate the difference between the proportions of firstborn or only children in the populations from which these samples were drawn. Use a 90% confidence interval and interpret your results.

An increase in the rate of consumer savings is frequently tied to a lack of confidence in the economy and is said to be an indicator of a recessional tendency in the economy. A random sampling of n=200 savings accounts in a local community showed a mean increase in savings account values of 7.2% over the past 12 months, with a standard deviation of 5.6%. Estimate the mean percent increase in savings account values over the past 12 months for depositors in the community. Find the margin of error for your estimate.