Stats hw 8.1-8.3

Suppose you are told that a 95% confidence interval for the average price of a gallon of regular gasoline in your state is from $3.28 to $4.50. Use the fact that the confidence interval for the mean is in the form x − E to x + E to compute the sample mean and the maximal margin of error E. (Round your answers to two decimal places.) x = $ E = $

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Suppose you want to test the claim that a population mean equals 40. (a) State the null hypothesis. (b) State the alternate hypothesis if you have no information regarding how the population mean might differ from 40. (c) State the alternative hypothesis if you believe (based on experience or past studies) that the population mean may exceed 40. (d) State the alternative hypothesis if you believe (based on experience or past studies) that the population mean may be less than 40.

H0: μ = 40 H1: μ ≠ 40 H1: μ > 40 H1: μ < 40

To test μ for an x distribution that is mound-shaped using sample size n ≥ 30, how do you decide whether to use the normal or Student's t distribution?

If σ is known, use the standard normal distribution. If σ is unknown, use the Student's t distribution with n - 1 degrees of freedom.

If we fail to reject (i.e., "accept") the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer.

No, it suggests that the evidence is not sufficient to merit rejecting the null hypothesis.

If we reject the null hypothesis, does this mean that we have proved it to be false beyond all doubt? Explain your answer.

No, the test was conducted with a risk of a type I error.

For the same sample data and null hypothesis, how does the P-value for a two-tailed test of μ compare to that for a one-tailed test?

The P-value for a two-tailed test is twice the P-value for a one-tailed test.

To use the normal distribution to test a proportion p, the conditions np > 5 and nq > 5 must be satisfied. Does the value of p come from H0, or is it estimated by using p̂ from the sample?

The value of p comes from H0.

A random sample of 25 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 7 and the sample standard deviation is 2. Use a level of significance of 0.05 to conduct a two-tailed test of the claim that the population mean is 6.5. (a) Is it appropriate to use a Student's t distribution? Explain. How many degrees of freedom do we use? (b) What are the hypotheses? (c) Compute the t value of the sample test statistic. (Round your answer to three decimal places.) (d) Estimate the P-value for the test. (e) Do we reject or fail to reject H0? (f) Interpret the results.

a. Yes, because the x distribution is mound-shaped and symmetric and σ is unknown. 24 H0: μ = 6.5; H1: μ ≠ 6.5 c. 1.250 d. 0.100 < P-value < 0.250 e. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. f. There is insufficient evidence at the 0.05 level to reject the null hypothesis.

Anystate Auto Insurance Company took a random sample of 356 insurance claims paid out during a 1-year period. The average claim paid was $1590. Assume σ = $232. a.) Find a 0.90 confidence interval for the mean claim payment. (Round your answers to two decimal places.) lower limit upper limit b.) Find a 0.99 confidence interval for the mean claim payment. (Round your answers to two decimal places.) lower limit upper limit

a.) 1569.77 1610.23 (1.645)(232 / (sqrt 356)) = 20.22688 1590 - 20.22688 = 1569.77 1590 + 20.22688 = 1610.23 b.) 1558.28 1621.72 (2.58)(232 / (sqrt 356)) = 31.723 1590 - 31.723 = 1558.28 1590 + 31.723 = 1621.72

The personnel office at a large electronics firm regularly schedules job interviews and maintains records of the interviews. From the past records, they have found that the length of a first interview is normally distributed, with mean μ = 37 minutes and standard deviation σ = 8 minutes. (Round your answers to four decimal places.) (a) What is the probability that a first interview will last 40 minutes or longer? (b) Seventeen first interviews are usually scheduled per day. What is the probability that the average length of time for the seventeen interviews will be 40 minutes or longer?

a.) .3520 (40-37) / (8 / (sqrt 1)) = .375 z score .38 = .64803 1 - .64803 = .35197 = .3520 b.) .0606 (40-37) / (8 / (sqrt 17)) = 1.55 z score 1.55 = .93943 1-.93943 = .06057 = .0606

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.Santa Fe black-on-white is a type of pottery commonly found at archaeological excavations at a certain monument. At one excavation site a sample of 602 potsherds was found, of which 350 were identified as Santa Fe black-on-white. (a) Let p represent the proportion of Santa Fe black-on-white potsherds at the excavation site. Find a point estimate for p. (Round your answer to four decimal places.) (b) Find a 95% confidence interval for p. (Round your answers to three decimal places.) lower limit upper limit Give a brief statement of the meaning of the confidence interval. (c) Do you think that np > 5 and nq > 5 are satisfied for this problem? Explain why this would be an important consideration.

a.) .5814 350/602 = .5814

A new muscle relaxant is available. Researchers from the firm developing the relaxant have done studies that indicate that the time lapse between administration of the drug and beginning effects of the drug is normally distributed, with mean μ = 38 minutes and standard deviation σ = 5 minutes. (a) The drug is administered to one patient selected at random. What is the probability that the time it takes to go into effect is 35 minutes or less? (Round your answer to four decimal places.) (b) The drug is administered to a random sample of 10 patients. What is the probability that the average time before it is effective for all 10 patients is 35 minutes or less? (Round your answer to four decimal places.) (c) Comment on the differences of the results in parts (a) and (b).

http://web2.slc.qc.ca/mh/QM/Review%20Exercises%20-%20Solutions.pdf ^work and explanation (Question 13 page 9) .2743 .0287 smaller, standard deviation, smaller

Weatherwise is a magazine published by the American Meteorological Society. One issue gives a rating system used to classify Nor'easter storms that frequently hit New England and can cause much damage near the ocean. A severe storm has an average peak wave height of μ = 16.4 feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. Peak wave heights are usually measured from land (using binoculars) off fixed cement piers. Suppose that a reading of 37 waves showed an average wave height of x = 16.9 feet. Previous studies of severe storms indicate that σ = 3.5 feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Estimate the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (e) Interpret your conclusion in the context of the application.

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Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of "good," socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the P/E, or price-to-earnings ratio. High P/E ratios may indicate a stock is overpriced. For the S&P Stock Index of all major stocks, the mean P/E ratio is μ = 19.4. A random sample of 36 "socially conscious" stocks gave a P/E ratio sample mean of x = 17.8, with sample standard deviation s = 5.4. Does this indicate that the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the S&P Stock Index? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Estimate the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (e) Interpret your conclusion in the context of the application.

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A random sample of 40 binomial trials resulted in 16 successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of significance of 0.05. (a) Can a normal distribution be used for the p̂ distribution? Explain. (b) State the hypotheses. (c) Compute p̂. Compute the corresponding standardized sample test statistic. (Round your answer to two decimal places.) (d) Find the P-value of the test statistic. (Round your answer to four decimal places.) (e) Do you reject or fail to reject H0? Explain. (f) What do the results tell you?

q. 19 watch it master it a. Yes, np and nq are both greater than 5. b. H0: p = 0.5; H1: p ≠ 0.5 c. 0.4 -1.26 d. 0.2076 e. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. f. The sample p̂ value based on 40 trials is not sufficiently different from 0.50 to justify rejecting H0 for α = 0.05.

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 38 arrests last month, 24 were of males aged 15 to 34 years. Use a 10% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (e) Interpret your conclusion in the context of the application.

q. 21 watch it a.) .1 H0: p = 0.7; H1: p ≠ 0.7 b.) The standard normal, since np > 5 and nq > 5. -.92 c.) .3576 d.) At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. e.) There is insufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.

What is your favorite color? A large survey of countries, including the United States, China, Russia, France, Turkey, Kenya, and others, indicated that most people prefer the color blue. In fact, about 24% of the population claim blue as their favorite color.† Suppose a random sample of n = 57 college students were surveyed and r = 9 of them said that blue is their favorite color. Does this information imply that the color preference of all college students is different (either way) from that of the general population? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (e) Interpret your conclusion in the context of the application.

q. 22 watch it a.) 0.05 H0: p = 0.24; H1: p ≠ 0.24 b.) The standard normal, since np > 5 and nq > 5. -1.45 c.) 0.1467 d.) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. e.) There is insufficient evidence at the 0.05 level to conclude that the true proportion of college students favoring the color blue differs from 0.24.

The average annual miles driven per vehicle in the United States is 11.1 thousand miles, with σ ≈ 600 miles. Suppose that a random sample of 26 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.8 thousand miles. Does this indicate that the average miles driven per vehicle in Chicago is different from (higher or lower than) the national average? Use a 0.05 level of significance. What are we testing in this problem? (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (e) Interpret your conclusion in the context of the application.

review q 17 watch it single mean .05 H0: μ = 11.1; H1: μ ≠ 11.1 The standard normal, since we assume that x has a normal distribution with known σ. -2.55 0.010 < P-value < 0.050

Professor Jennings claims that only 35% of the students at Flora College work while attending school. Dean Renata thinks that the professor has underestimated the number of students with part-time or full-time jobs. A random sample of 77 students shows that 37 have jobs. Do the data indicate that more than 35% of the students have jobs? Use a 5% level of significance. What are we testing in this problem? (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (e) Interpret your conclusion in the context of the application.

review q 18 watch it single proportion .05 H0: p = 0.35; H1: p > 0.35 The standard normal, since np > 5 and nq > 5. 2.4 0.005 < P-value < 0.025

How much do wild mountain lions weigh? Adult wild mountain lions (18 months or older) captured and released for the first time in the San Andres Mountains gave the following weights (pounds): 73,107,132,122,60,64 Assume that the population of x values has an approximately normal distribution. (a) Use a calculator with mean and sample standard deviation keys to find the sample mean weight x and sample standard deviation s. (Round your answers to one decimal place.) (b) Find a 75% confidence interval for the population average weight μ of all adult mountain lions in the specified region. (Round your answers to one decimal place.) lower limit upper limit

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The body weight of a healthy 3-month-old colt should be about μ = 62 kg. (a) If you want to set up a statistical test to challenge the claim that μ = 62 kg, what would you use for the null hypothesis (b) In Nevada, there are many herds of wild horses. Suppose you want to test the claim that the average weight of a wild Nevada colt (3 months old) is less than 62 kg. What would you use for the alternate hypothesis H1? (c) Suppose you want to test the claim that the average weight of such a wild colt is greater than 62 kg. What would you use for the alternate hypothesis? (d) Suppose you want to test the claim that the average weight of such a wild colt is different from 62 kg. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), respectively, would the area corresponding to the P-value be on the left, on the right, or on both sides of the mean?

μ = 62 kg μ < 62 kg μ > 62 kg μ ≠ 62 kg left; right; both