BUSI 230 Review Exam 3 Inferential Statistics solutions complete answers

BUSI 230 Review Exam 3 Inferential Statistics solutions complete answers 

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TV sets: According to the Nielsen Company, the mean number of TV sets in a U.S. household in 2013 was 2.24. Assume the standard deviation is 1.3. A sample of 80 households is drawn. 

(a) What is the probability that the sample mean number of TV sets is greater than 2? Round your answer to four decimal places.

(b) What is the probability that the sample mean number of TV sets is between 2.5 and 3? Round your answer to four decimal places.

(c) Find the 80 percentile of the sample mean. Round your answer to two decimal places.

(d) Would it be unusual for the sample mean to be less than 2? Round your answer to four decimal places.

(e) Do you think it would be unusual for an individual household to have fewer than 2 TV sets? Explain. Assume the population is approximately normal. Round your answer to four decimal places.

 

Blood pressure: High blood pressure has been identified as a risk factor for heart attacks and strokes. The proportion of U.S. adults with high blood pressure is 0.40. A sample of 19 U.S. adults is chosen. Use Excel as needed.

(a) Is it appropriate to use the normal approximation to find the probability that more than 43% of the people in the sample have high blood pressure? If so, find the probability. If not, explain why not.

(b) A new sample of 79 adults is drawn. Find the probability that more than 38% of the people in this sample have high blood pressure. Round the answer to at least four decimal places.

(c) Find the probability that the proportion of individuals in the sample of  who have high blood pressure is between 0.25 and 0.33. Round the answer to at least four decimal places.

(d) Find the probability that less than 28% of the people in the sample of  have high blood pressure. Round the answer to at least four decimal places.

(e) Would it be unusual if more than 34% of the individuals in the sample of  had high blood pressure? Round the answer to at least four decimal places.

 

How many computers? In a simple random sample of 140 households, the sample mean number of personal computers was 1.07. Assume the population standard deviation is 0.72.

(a) Construct a 95% confidence interval for the mean number of personal computers. Round the answer to at least two decimal places.

(b) If the sample size were  rather than , would the margin of error be larger or smaller than the result in part (a)? Explain.

(c) If the confidence levels were  rather than , would the margin of error be larger or smaller than the result in part (a)? Explain.

(d) Based on the confidence interval constructed in part (a), is it likely that the mean number of personal computers is less than ?

 

Let's go to the movies: A random sample of 45 Foreign Language movies made since 2,000 had a mean length of 111.6 minutes, with a standard deviation of 14.3 minutes.

(a) Construct a 99% confidence interval for the true mean length of all Foreign Language movies made since . Round the answers to one decimal place.

(b) As of November , three Spider-Man movies have been released, and their mean length is  minutes. Someone claims that the mean length of Spider-Man movies is actually less than the mean length of all Foreign Language movies. Does the confidence interval contradict this claim?

 

Eat your cereal: Boxes of cereal are labeled as containing 14 ounces. Following are the weights, in ounces, as of a sample of 16 boxes. It is reasonable to assume that the population is approximately normal.

(a) Construct a 95% confidence interval for the mean weight. Round the answers to three decimal places.

(b) The quality control manager is concerned that the mean weight is actually less than  ounces. Based on the confidence interval, is there a reason to be concerned? Explain.

 

Internet service: An Internet service provider sampled 555 customers, and finds that 78 of them experienced an interruption in high-speed service during the previous month.

(a) Find a point estimate for the population proportion of all customers who experienced an interruption. Round the answer to at least three decimal places.

(b) Construct a 99% confidence interval for the proportion of all customers who experienced an interruption. Round the answers to at least three decimal places.

(c) The company's quality control manager claims that no more than  of its customers experienced an interruption during the previous month. Does the confidence interval contradict this claim? Explain.

 

Reading proficiency: An educator wants to construct a 99.9% confidence interval for the proportion of elementary school children in Colorado who are proficient in reading.

(a) The results of a recent statewide test suggested that the proportion is 0.68. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.04?

(b) Estimate the sample size needed if no estimate of p is available.

(c) If the educator wanted to estimate the proportion in the entire United States rather than in Colorado, would the necessary size be larger, smaller, or about the same? Explain.

 

Determine whether the alternate hypothesis is left-tailed, right-tailed, or two-tailed.

 

Determine whether the outcome is a Type I error, a Type II error, or a correct decision.

 

Coffee: The mean caffeine content per cup of regular coffee served at a certain coffee shop is supposed to be different than . A test is made of  versus . The null hypothesis is rejected. State an appropriate conclusion.

 

IQ: Scores on a certain IQ test are known to have a mean of 100. A random sample of 54 students attend a series of coaching classes before taking the test. Let  be the population mean IQ score that would occur if every student took the coaching classes. The classes are successful if u>100. A test is made of the hypotheses H0:u=100 versus H1:u>100. Consider three possible conclusions: (i) The classes are successful. (ii) The classes are not successful. (iii) The classes might not be successful.

If P=0.05, the result is statistically significant at the =0.02 level.

 

SAT scores: The College Board reports that in  the mean score on the math SAT was 527 and the population standard deviation was =112. A random sample of 20 students who took the test in  had a mean score of 538. Following is a dotplot of the 20 scores.

(a) Are the assumptions for a hypothesis test satisfied?

(b) Perform a hypothesis test to determine whether you can conclude that the mean score in  differs from the mean score in . Assume the population standard deviation is =112. Use the =0.10 level of significance and the -value method with the TI-84 Plus calculator.

 

Height and age: Are older men shorter than younger men? According to a national report, the mean height for U.S. men is 69.4 inches. In a sample of 301 men between the ages of 60 and 69, the mean height was =69.2 inches. Public health officials want to determine whether the mean height  for older men is less than the mean height of all adult men. Assume the population standard deviation to be =2.96. Use the =0.10 level of significance and the P-value method with the TI-84 calculator.

 

College tuition: The mean annual tuition and fees for a sample of 26 private colleges in California was $38,200 with a standard deviation of $7,000. A dotplot shows that it is reasonable to assume that the population is approximately normal. Can you conclude that the mean tuition and fees for private institutions in California differs from $35,000? Use the =0.10 level of significance and the P-value method with the TI-84 Plus calculator.

 

Good credit: The Fair Isaac Corporation (FICO) credit score is used by banks and other lenders to determine whether someone is a good credit risk. Scores range from 300 to 850, with a score of 720 or more indicating that a person is a very good credit risk. An economist wants to determine whether the mean FICO score is lower than the cutoff of 720. She finds that a random sample of 55 people had a mean FICO score of 695 with a standard deviation of 65. Can the economist conclude that the mean FICO score is less than 720? Use the =0.05 level of significance and the P-value method with the TI-84 Plus calculator.

 

Tweet tweet: An article reported that 73% of companies have Twitter accounts. An economist thinks the percentage is higher at technology companies. She samples 70 technology companies and finds that 46 of them have Twitter accounts. Can she conclude that less than 73% of technology companies have Twitter accounts? Use the =0.05 level of significance and the P-value method and Excel.

 

SAT scores: Assume that in a given year the mean mathematics SAT score was 605, and the standard deviation was 136. A sample of 76 scores is chosen. Use Excel.

(a) What is the probability that the sample mean score is less than 589? Round the answer to at least four decimal places.

(b) What is the probability that the sample mean score is between 575 and 610? Round the answer to at least four decimal places.

(c) Find the 85 percentile of the sample mean. Round the answer to at least two decimal places.

(d) Would it be unusual if the sample mean were greater than 620? Round the answer to at least four decimal places.

(e) Do you think it would be unusual for an individual to get a score greater than 620? Explain. Assume the variable is normally distributed. Round the answer to at least four decimal places.

 

Smartphones: A poll agency reports that 24% of teenagers aged 12-17 own smartphones. A random sample of 75 teenagers is drawn. Round your answers to at least four decimal places as needed.

(a) Find the mean .

(b) Find the standard deviation .

(c) Find the probability that more than 26% of the sampled teenagers own a smartphone.

(d) Find the probability that the proportion of the sampled teenagers who own a smartphone is between 0.21 and 0.30.

(e) Find the probability that less than 30% of sampled teenagers own smartphones.

(f) Would it be unusual if less than 15% of the sampled teenagers owned smartphones?

 

Babies: According to a recent report, a sample of 300 one-year-old baby boys in the United States had a mean weight of 25.5 pounds. Assume the population standard deviation is 5.3 pounds.

(a) Construct a 99.9% confidence interval for the mean weight of all one-year-old baby boys in the United States. Round the answer to at least one decimal place.

(b) Should this confidence interval be used to estimate the mean weight of all one-year-old babies in the United States? Explain.

(c) Based on the confidence interval constructed in part (a), is it likely that the mean weight of all one-year-old boys is greater than  pounds?

 

Hip surgery: In a sample of 126 hip surgeries of a certain type, the average surgery time was 137.5 minutes, with a standard deviation of 22.9 minutes.

(a) Construct a 90% confidence interval for the mean surgery time for this procedure. Round your answers to one decimal place

(b) If a  confidence interval were constructed with these data, would it be wider or narrower than the interval constructed in part (a)? Explain.

 

Volunteering: The General Social Survey asked 1296 people whether they performed any volunteer work during the past year. A total of 531 people said they did.

(a) Find a point estimate for the population proportion of people who performed volunteer work in the past year. Round the answer to at least three decimal places.

(b) Construct a 90% confidence interval for the proportion of people who performed volunteer work in the past year. Round the answer to at least three decimal places.

(c) A sociologist states that  of Americans perform volunteer work in a given year. Does the confidence interval contradict this statement? Explain.

 

Surgical complications: A medical researcher wants to construct a 98% confidence interval for the proportion of knee replacement surgeries that result in complications.

(a) An article in a medical journal suggested that approximately 12% of such operations result in complications. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.06?

(b) Estimate the sample size needed if no estimate of p is available.

 

Type I error: A company that manufactures steel wires guarantees that the mean breaking strength (in kilonewtons) of the wires is greater than 50. They measure the strengths for a sample of wires and test  versus .

If a Type I error is made, what conclusion will be drawn regarding the mean breaking strength?

If a Type II error is made, what conclusion will be drawn regarding the mean breaking strength?

 

Are you smarter than a second-grader? A random sample of 58 second-graders in a certain school district are given a standardized mathematics skills test. The sample mean score is x=47. Assume the standard deviation of test scores is =15. The nationwide average score on this test is 50. The school superintendent wants to know whether the second-graders in her school district have weaker math skills than the nationwide average. Use the =0.10 level of significance and the P-value method with the TI-84 calculator.

 

Big babies: The National Health Statistics Reports described a study in which a sample of 305 one-year-old baby boys were weighed. Their mean weight was 25.6 pounds with standard deviation 5.3 pounds. A pediatrician claims that the mean weight of one-year-old boys differs from 25 pounds. Do the data provide convincing evidence that the pediatrician's claim is true? Use the =0.01 level of significance and the P-value method with the TI-84 Plus calculator.

 

Confidence in banks: A poll conducted asked a random sample of 1276 adults in the United States how much confidence they had in banks and other financial institutions. A total of 150 adults said that they had a great deal of confidence. An economist claims that less than 14% of U.S. adults have a great deal of confidence in banks. Can you conclude that the economist's claim is true? Use both =0.01 and =0.05 levels of significance and the -value method with the TI-84 Plus calculator.

 

Lifetime of electronics: In a simple random sample of 100 electronic components produced by a certain method, the mean lifetime was 125 hours. Assume that component lifetimes are normally distributed with population standard deviation =20 hours. Round the critical value to no less than three decimal places.

(a) Construct a 98% confidence interval for the mean battery life. Round the answer to the nearest whole number.

(b) Find the sample size needed so that a 99.9% confidence interval will have a margin of error of 4.

 

SAT scores:The mean mathematics SAT score was 524, and the standard deviation was 117. A sample of 68 scores is chosen. Use the Cumulative Normal Distribution Table if needed.

(a) What is the probability that the sample mean score is less than 512? Round the answer to at least four decimal places.

(b) What is the probability that the sample mean score is between 485 and 529? Round the answer to at least four decimal places.

(c) Find the 10 percentile of the sample mean. Round the answer to at least two decimal places.

(d) Would it be unusual if the sample mean were greater than 529? Round the answer to at least four decimal places.

(e) Do you think it would be unusual for an individual to get a score greater than 529? Explain. Assume the variable is normally distributed. Round the answer to at least four decimal places.

 

Student loans: The Institute for College Access and Success reported that 65% of college students in a recent year graduated with student loan debt. A random sample of 75 graduates is drawn. Use Cumulative Normal Distribution Table as needed. Round your answers to at least four decimal places if necessary.

(a) Find the mean .

(b) Find the standard deviation .

(c)Find the probability that less than 52% of the people in the sample were in debt.

(d)Find the probability that between 55% and 75% of the people in the sample were in debt.

(e)Find the probability that more than 67% of the people in the sample were in debt.

(f)Would it be unusual if less than 51% of people in the sample were in debt?

 

Efficient manufacturing: Efficiency experts study the processes used to manufacture items in order to make them as efficient as possible. One of the steps used to manufacture a metal clamp involves the drilling of three holes. In a sample of 50 clamps, the mean time to complete this step was 40.4 seconds. Assume that the population standard deviation is =7 seconds. Round the critical value to no less than three decimal places.

(a) Construct a 98% confidence interval for the mean time needed to complete this step. Round the answer to at least one decimal place.

(b) Find the sample size needed so that a 99% confidence interval will have margin of error of 1.5.