Liberty University BUSI 230 review and exam 3 complete solutions answers and more!

Liberty University BUSI 230 review and exam 3 complete solutions answers and more!

In general, a large value for a t statistic (far from zero) is an indication that the sample data are not consistent with the null hypothesis.

If all other factors are held constant, increasing the sample size will do the following.

A Type I error is defined as the following.

The t distribution is symmetrical and has a mean of zero.

Rejecting the null hypothesis means that the sample outcome is very unlikely to have occurred if H0 is true.

If a specific sample leads to rejecting the null hypothesis with α = .05, then the same sample would certainly lead to rejecting the null hypothesis if α were changed to .01.

A Type II error is defined as the following.

There is always a possibility that the decision reached in a hypothesis test is incorrect.

In most research situations, the goal of a hypothesis test is to reject the null hypothesis.

Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a certain article, the mean of the x distribution is about $47 and the estimated standard deviation is about $9.

(a) Consider a random sample of n = 80 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?

Is it necessary to make any assumption about the x distribution? Explain your answer.

(b) What is the probability that x is between $45 and $49? (Round your answer to four decimal places.)

(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $45 and $49? (Round your answer to four decimal places.)

(d) In part (b), we used x, the average amount spent, computed for 80 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen?

Answer

In this example, x is a much more predictable or reliable statistic than x. Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

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).

The probability in part (b) is     part (a) because the     is     for the x distribution.

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 = 36 minutes and standard deviation σ = 6 minutes. (Round your answers to four decimal places.)

(a) What is the probability that a first interview will last 40 minutes or longer?

(b) Eighteen first interviews are usually scheduled per day. What is the probability that the average length of time for the eighteen interviews will be 40 minutes or longer?

How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds, by Wirth and Young (Random House), claims that the air in the crown should be an average of 100°C for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly 100°C. What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 57 readings (for a balloon in equilibrium) gave a mean temperature of x = 97°C. For this balloon, σ ≈ 16°C.

(a) Compute a 95% confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (Round your answers to one decimal place.)

(b) If the average temperature in the crown of the balloon goes above the high end of your confidence interval, do you expect that the balloon will go up or down? Explain.

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):

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.)

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 592 potsherds was found, of which 361 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.)

Give a brief statement of the meaning of the confidence interval.

Answer

(c) Do you think that np > 5 and nq > 5 are satisfied for this problem? Explain why this would be an important consideration.

Suppose you are told that a 95% confidence interval for the average price of a gallon of regular gasoline in your state is from $3.46 to $4.41. 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.)

Anystate Auto Insurance Company took a random sample of 386 insurance claims paid out during a 1-year period. The average claim paid was $1540. Assume σ = $264.

Find a 0.90 confidence interval for the mean claim payment. (Round your answers to two decimal places.)

margin of error

Find a 0.99 confidence interval for the mean claim payment. (Round your answers to two decimal places.)

Three experiments investigating the relation between need for cognitive closure and persuasion were performed. Part of the study involved administering a "need for closure scale" to a group of students enrolled in an introductory psychology course. The "need for closure scale" has scores ranging from 101 to 201. For the 71 students in the highest quartile of the distribution, the mean score was x = 176.50. Assume a population standard deviation of σ = 7.81. These students were all classified as high on their need for closure. Assume that the 71 students represent a random sample of all students who are classified as high on their need for closure. Find a 95% confidence interval for the population mean score μ on the "need for closure scale" for all students with a high need for closure. (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

The Wind Mountain archaeological site is located in southwestern New Mexico. Wind Mountain was home to an ancient culture of prehistoric Native Americans called Anasazi. A random sample of excavations at Wind Mountain gave the following depths (in centimeters) from present-day surface grade to the location of significant archaeological artifacts†.

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to one decimal place.)

x =

s =

(b) Compute a 95% confidence interval for the mean depth μ at which archaeological artifacts from the Wind Mountain excavation site can be found. (Round your answers to one decimal place.)

lower limit

upper limit

A research group conducted an extensive survey of 2994 wage and salaried workers on issues ranging from relationships with their bosses to household chores. The data were gathered through hour-long telephone interviews with a nationally representative sample. In response to the question, "What does success mean to you?" 1635 responded, "Personal satisfaction from doing a good job." Let p be the population proportion of all wage and salaried workers who would respond the same way to the stated question. Find a 90% confidence interval for p. (Round your answers to three decimal places.)

lower limit

upper limit

Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archaeological site. At one excavation, a sample of 177 potsherds indicated that 80 were of the three-circle, red-on-white pattern.

(a) Find a point estimate p̂ for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern. (Round your answer to four decimal places.)

(b) Compute a 95% confidence interval for the population proportion p of all ceramic potsherds with this distinctive pattern found at the site. (Round your answers to three decimal places.)

Consumer Reports stated that the mean time for a Chrysler Concorde to go from 0 to 60 miles per hour was 8.7 seconds.

(a) If you want to set up a statistical test to challenge the claim of 8.7 seconds, what would you use for the null hypothesis?

Answer

(b) The town of Leadville, Colorado, has an elevation over 10,000 feet. Suppose you wanted to test the claim that the average time to accelerate from 0 to 60 miles per hour is longer in Leadville (because of less oxygen). What would you use for the alternate hypothesis?

Answer

(c) Suppose you made an engine modification and you think the average time to accelerate from 0 to 60 miles per hour is reduced. What would you use for the alternate hypothesis?

Answer

(d) For each of the tests in parts (b) and (c), would the P-value area be on the left, on the right, or on both sides of the mean?

The price to earnings ratio (P/E) is an important tool in financial work. A random sample of 14 large U.S. banks (J. P. Morgan, Bank of America, and others) gave the following P/E ratios.†

The sample mean is x ≈ 17.1. Generally speaking, a low P/E ratio indicates a "value" or bargain stock. Suppose a recent copy of a magazine indicated that the P/E ratio of a certain stock index is μ = 18. Let x be a random variable representing the P/E ratio of all large U.S. bank stocks. We assume that x has a normal distribution and σ = 4.7. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 18? Use α = 0.01.

(a) What is the level of significance?

Answer

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

Answer

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

Answer

Compute the z value of the sample test statistic. (Round your answer to two decimal places.)

Answer

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Answer

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?

Answer

(e) State your conclusion in the context of the application.

A random sample of 51 adult coyotes in a region of northern Minnesota showed the average age to be x = 2.01 years, with sample standard deviation s = 0.76 years. However, it is thought that the overall population mean age of coyotes is μ = 1.75. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use α = 0.01.

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

Answer

What is the value of the sample test statistic? (Round your answer to three decimal places.)

Answer

(c) Estimate the P-value.

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

The U.S. Department of Transportation, National Highway Traffic Safety Administration, reported that 77% of all fatally injured automobile drivers were intoxicated. A random sample of 52 records of automobile driver fatalities in a certain county showed that 34 involved an intoxicated driver. Do these data indicate that the population proportion of driver fatalities related to alcohol is less than 77% in Kit Carson County? Use α = 0.10.

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use?

Answer

What is the value of the sample test statistic? (Round your answer to two decimal places.)

0.65384615

Answer

(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

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 41 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.9 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?

Answer

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use? What assumptions are you making?

Answer

What is the value of the sample test statistic? (Round your answer to two decimal places.)

Answer

(c) Find (or estimate) the P-value.

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

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 84 students shows that 40 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?

Answer

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use? What assumptions are you making?

Answer

What is the value of the sample test statistic? (Round your answer to two decimal places.)

0.47619048

Answer

(c) Find (or estimate) the P-value.

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

The Toylot company makes an electric train with a motor that it claims will draw an average of only 0.8 ampere (A) under a normal load. A sample of nine motors was tested, and it was found that the mean current was x = 1.32 A, with a sample standard deviation of s = 0.42 A. Do the data indicate that the Toylot claim of 0.8 A is too low? (Use a 1% level of significance.)

What are we testing in this problem?

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use? What assumptions are you making?

Answer

What is the value of the sample test statistic? (Round your answer to three decimal places.)

Answer

(c) Find (or estimate) the P-value.

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

A hospital reported that the normal death rate for patients with extensive burns (more than 40% of skin area) has been significantly reduced by the use of new fluid plasma compresses. Before the new treatment, the mortality rate for extensive burn patients was about 60%. Using the new compresses, the hospital found that only 43 of 93 patients with extensive burns died. Use a 1% level of significance to test the claim that the mortality rate has dropped.

What are we testing in this problem?

Answer

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use? What assumptions are you making?

Answer

What is the value of the sample test statistic? (Round your answer to two decimal places.)

0.46236559

Answer

(c) Find (or estimate) the P-value.

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna).† Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were as follows.

The sample mean is x = 3.75 grams. Let x be a random variable representing weights of hummingbirds in this part of the Grand Canyon. We assume that x has a normal distribution and σ = 0.96 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.80 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.80 grams? Use α = 0.10.

(a) What is the level of significance?

Answer

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

Answer

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

Answer

Compute the z value of the sample test statistic. (Round your answer to two decimal places.)

Answer

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Answer

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?

Answer

(e) State your conclusion in the context of the application.

Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 43 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 8.00 ml/kg for the distribution of blood plasma.

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

(c) Interpret your results in the context of this problem.

Answer

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.50 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)

The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to the nearest whole number.)

(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

Women athletes at the a certain university have a long-term graduation rate of 67%. Over the past several years, a random sample of 41 women athletes at the school showed that 22 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the university is now less than 67%? Use a 1% level of significance.

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use?

Answer

What is the value of the sample test statistic? (Round your answer to two decimal places.)

0.53658537

Answer

(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of 5180 permanent dwellings on an entire reservation showed that 1676 were traditional hogans.

(a) Let p be the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate for p. (Round your answer to four decimal places.)

Answer

(b) Find a 99% confidence interval for p. (Round your answer to three decimal places.)

Give a brief interpretation of the confidence interval.

Answer

(c) Do you think that np > 5 and nq > 5 are satisfied for this problem? Explain why this would be an important consideration.

Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is μ = 7.4.† A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 41 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.7 with sample standard deviation s = 3.4. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood.

(a) What is the level of significance?

Answer

State the null and alternate hypotheses.

Answer

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

Answer

What is the value of the sample test statistic? (Round your answer to three decimal places.)

Answer

(c) Estimate the P-value.

Answer

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?

Answer

(e) Interpret your conclusion in the context of the application.

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 71 inches and standard deviation 1 inches.

(a) What is the probability that an 18-year-old man selected at random is between 70 and 72 inches tall? (Round your answer to four decimal places.)

(b) If a random sample of thirty 18-year-old men is selected, what is the probability that the mean height x is between 70 and 72 inches? (Round your answer to four decimal places.)

(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

Weatherwise magazine is published in association with the American Meteorological Society. Volume 46, Number 6 has a rating system to classify Nor'easter storms that frequently hit New England states and can cause much damage near the ocean coast. A severe storm has an average peak wave height of 16.4 feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating.

(a) Let us say that we want to set up a statistical test to see if the wave action (i.e., height) is dying down or getting worse. What would be the null hypothesis regarding average wave height?

Answer

(b) If you wanted to test the hypothesis that the storm is getting worse, what would you use for the alternate hypothesis?

Answer

(c) If you wanted to test the hypothesis that the waves are dying down, what would you use for the alternate hypothesis?

Answer

(d) Suppose you do not know if the storm is getting worse or dying out. You just want to test the hypothesis that the average wave height is different (either higher or lower) from the severe storm class rating. What would you use for the alternate hypothesis?

Answer

(e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the P-value be on the left, on the right, or on both sides of the mean?