Examples PDF

Example 1 A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips etc). While none had more than three of these defects, 7% had three, 11% had two, and 21% had one defect). a) Find the expected number of appearance defects in a new...

Example 1 A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips etc). While none had more than three of these defects, 7% had three, 11% had two, and 21% had one defect). a) Find the expected number of appearance defects in a new car. b) What is the standard deviation?  Example 2 Your company bids for two contracts. You believe the probability that you get contract #1 is 0.8. If you get contract #1, the probability that you also get contract #2 will be 0.2, and if you do not get contract #1, the probability that you get contract #2 will be 0.3. a) Are the outcomes of the two contract bids independent? b) Find the probability that you get both contracts c) Find the probability that you get neither contract d) Let X be the number of contracts you get. Find the expected value and standard deviation of X  Example 3 A grocery supplier believes that the mean number of broken eggs per dozen is 0.6, with a standard deviation of 0.5. You buy three dozen eggs without checking them. a) How many broken eggs do you expect to get? b) What is the standard deviation c) Is it necessary to assume the cartons of eggs are independent? Why?  Example 4 An insurance company estimates that it should make an annual profit of $150 on each homeowner’s policy written, with a standard deviation of $6000. a) Why is the standard deviation so large? b) If the company writes only two of these policies, what are the mean and standard deviation of the annual profit? c) If the company writes 1000 of these policies, what are the mean and standard deviation of the annual profit? d) What circumstances could violate the assumption of independence of the policies?  Example 5 A bicycle shop plans to offer 2 specially priced children’s models at a sidewalk sale. The basic model will return a profit of $120 and the deluxe model $150. Past experience indicates that sales of the basic model will have a mean of 5.4 bikes with a standard deviation of 1.2, and sales of the deluxe model will have a mean of 3.2 bikes with a standard deviation of 0.8 bikes. The cost of setting up for the sidewalk sale is $200. a) Define random variables and use them to express the bicycle shop’s net profit. b) What is the mean of the net profit? c) What is the standard deviation of the net profit? d) Do you need to make any assumptions in calculating the mean? How about the standard deviation?  Let Y = the number of dropped calls experienced per day by customers of a particular cell phone provider. This random variable is A. measurable. B. discrete. C. continuous. D. a failure.  Suppose that in the previous example the values of Y are 0, 1, 2, or 3. Also suppose that the probability associated with each of these values is.55,.30,.10 and.05, respectively. What is the expected number of dropped calls per day? A. 0 B. 0.5 C. 0.65  Suppose the probability of a dropped call is.02. If we want to determine the probability that 2 out of the next 20 calls placed on a cell will drop, what probability model should we use? A. Binomial B. Uniform C. Geometric D. Poisson  Let Y = the time it takes technical support staff to complete trouble shooting on a service call. This random variable is A. discrete. B. continuous. C. binomial. D. expected.  Which of the following is not a characteristic of the normal probability model? A. The normal random variable is discrete. B. The normal distribution is bell shaped. C. The sum of two independent normal random variables is also normal. D. The mean, median and mode are all equal for the normal distribution.  Suppose the time business students study for a statistics exam is normally distributed with a mean of 6 hours and standard deviation of 1.5 hours. What is the probability a student studies more than 8 hours? A. 1.33 B. 0.0918 C. 0.9082 D. 0.50  What conditions must be met to use the normal model as an approximation to the binomial? A. There are only two possible outcomes for each trial. B. The trials are independent. C. The expected number of successes and failures is each at least 10. D. All of the above  Example 1 An investment website can tell what devices are used to access the site. The site manager wonder whether they should enhance the facilities for trading via “smart phones’’ so they want to estimate the proportion of users who access the site that way. They draw a sample of 200 investors. Suppose that the true proportion of smart phone users is 36%. a) What would you expect the shape of the sampling distribution for the sample proportion to be? b) What would the mean and standard deviation of this sampling distribution be? c) If the sample size were increased to 400, would your answer change? d) What is the probability that the sample proportion of smart phone users is between 30% and 40%?  Example 2 The proportion of adult women in the US is approximately 50%.. A marketing survey telephones 400 people at random. a) Would you be surprised to find 53% women in the sample? b) Would you be surprised to find 41% women in the sample? c) Would you be surprised to find fewer than 160 women in the sample?  Example 3 The campus representative for Lens.com wants to know what percentage of students at a university currently wear contact lenses. Suppose that the the true proportion is 30% and we randomly choose 100 students. What is the probability that more than one third of the students in the sample wear contacts?  Example 4 An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them. a) Create a 95% confidence interval for the percentage of all auto accidents that involve teenage drivers. b) Explain what your interval means c) Explain what “95% confidence” means d) A politician urging tighter restrictions on drivers’ licenses issued to teens says, “In one of every five auto accidents, a teenager is behind the wheel”. Does your confidence interval support or contradict this statement? Explain  Example 5 From a survey of coworkers you find that 48% of 200 have already received this year’s flu vaccine. A 95% confidence interval is (.409,.551). a) How would the CI change if the sample size had been 800 instead of 200? b) How would the CI change if the confidence level had been 90% instead of 95%? c) How would the CI change if the confidence level had been 99% instead of 95%?  Example 6 Based on a survey, a market researcher estimated the percentage of adults between the ages of 21 and 39 who will see their television ads is 15% with a margin of error of about 3% with 95% confidence a) Explain what the margin of error means b) How can the market researcher decrease the margin of error?  Example 7 A state’s environmental agency worries that a large percentage of cars may be violating clear air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of 3% and 90% confidence. To gauge the size of the problem, the agency first picks 60 cars and finds 9 with faulty emission systems. How many should be sampled for a full investigation?  Example 1 A restaurant chain claims that 80% of its customers are satisfied with its service. If you choose a random sample of 9 customers what is the chance that a) exactly 8 customers are satisfied? b) more than 7 customers are satisfied? c) none are satisfied? d) At least one customer is not satisfied?  Example 2 Suppose a computer chip manufacturer rejects 2% of the chips produced because they fail presale testing. If 20 chips are randomly chosen, what is the chance that a) all of them are rejected? b) none of them are rejected? c) exactly 4 are rejected? d) at most two are rejected? e) not all of them are rejected?  Example 3 An e-commerce web site claims that 25% of people who visit the site make a purchase. Based on a random sample of 100 people who visit the site, what is the chance that a) exactly 40 will make a purchase? b) less than 40 will make a purchase?  Example 4 A production line produces AA batteries with a reliability rate of 95%. A sample of n = 200 batteries is selected. Find the probability that at least 195 of the batteries work.  Example 5 Toss a coin 100 times. What is the chance of seeing a) 52 or 53 Heads? b) over 55 Heads?  Example 1 For each of the following, write out the null and alternative hypothesis, being sure to state whether it is one-sided or two-sided. a) A company reports that last year 40% of their reports in accounting were on time. From a random sample this year, they want to know if the proportion has changed. b) Last year, 42% of the employees enrolled in at least one wellness class at the company’s site. Using a survey, they want to know if a greater percentage is planning to take a wellness class this year c) A political candidate wants to know from recent polls if she’s going to garner a majority of votes in the next week’s election d) Consumer Reports discovered that 20% of a certain computer model had warranty problems over the first three months. From a random sample, the manufacturer wants to know if a new model has improved that rate. e) A market researcher for a cola company decides to field test a new flavor soft drink, planning to market it only and only if he/she is sure that over 60% of the people like the flavor. f) Recently, 20% of cars of a certain model have need costly transmission work after being driven between 50000 and 100000 miles. The car manufacturer hopes that the redesign of a transmission component has solved this problem  Example 2 A consulting firm had predicted that 35% of the employees at a large firm would take advantage of a new Credit Union, but management is skeptical. They doubt that rate is that high. A survey of 300 employees shows that 138 of them are currently taking advantage of the Credit Union. a) What are the appropriate hypotheses? b) Find the standard deviation of the sample proportion based on the null hypothesis c) Find the z-statistic d) Does the z-statistic seem like a particularly large or small value?  Example 3 Which of the following are true? If false, explain briefly. a) A very high p-value is strong evidence that the null hypothesis is false b) A very low p-value proves that the null hypothesis is false. c) A high p-value shows that null hypothesis is true d) A p-value below 0.05 is always considered sufficient evidence to reject a null hypothesis  Example 4 A pharmaceutical company’s old antacid formula provided relief for 70% of the people who used it. The company tests a new formula to see if it is better and gets a p-value of.27. a) Is it reasonable to conclude that the new formula is more effective than the old one? b) B) Is it reasonable that the new formula and the old one are equally effective? Explain  Example 5 A mutual fund manager claims that at least 70% of the stocks she selects will increase in price over the next year. We examine a sample of 200 of her selection over the past three years. Test the appropriate hypotheses. Which conclusion is appropriate? Explain a) There is a 3% chance that the fund manager is correct  Example 5 b) There is a 97% chance that the fund manager is right c) There is a 3% chance that a random sample could produce the results we observed, so it’s reasonable to conclude that the fund manager is correct d) There is a 3% chance that a random sample could produce the results we observed if p=.7, so it’s reasonable to conclude that the fund manager is not correct e) There is a 3% chance that the null hypothesis is correct  Example 6 A magazine called Webzine is considering the launch of an online edition. The magazine plans to go ahead only if it’s convinced that more than 25% of current readers would subscribe. The magazine contacts a random sample of 500 current subscribers, and 150 of those surveyed expressed interest. a) Test the appropriate hypotheses b) Find the p-value c) Should the company launch the online edition? Explain  Example 1 A billing company that collects bills for doctors’ offices in the area is concerned that the percentage of bills paid by Medicare has risen. Historically, that percentage has been 31%. An examination of 8368 recent bills reveals that 32% of these bills are being paid by Medicare. Is there evidence that the company’s concern is legit? a) Write the appropriate hypotheses. b) Check the assumptions and conditions. c) Perform the test and find p-value at 5% level of significance.  Example 2 A start-up company is about to market a new computer printer. It decides to gamble by running commercials during the Super Bowl. The company hopes that name recognition will be worth the high cost of the ads. The goal of the company is that over 40% of the public recognize its brand name and associate it with computer equipment. The day after the game, a pollster contacts 420 randomly chosen adults and finds that 181 of them know that this company manufactures printers. Would you recommend that the company continue to advertise during the Super Bowl? Explain.  Example 3 A company hopes to improve customer satisfaction, setting as a goal of less than 5% negative complaints. A random sample of 400 customers found only 15 with complaints. Based on this sample, does the company reach its goal? Test the appropriate hypotheses at 5% level of significance using p-values, critical regions and confidence intervals and state your conclusion.  Example 4 Production managers on an assembly line must monitor the output to be sure that the level of defective products remain small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. a) Write null and alternative hypotheses for this problem. b) What are the Type I and Type II errors for this problem? c) Which type of error would the factory owner consider more serious? d) Which type of error might customers consider more serious?  Example 1 The average age of online consumers a few years ago was 23.3. As older individuals gain confidence with the Internet, it is believed that the average age has increased. We would like to test this belief. a) Write the appropriate hypotheses. b) We plan to test the hypotheses by selecting a random sample of 40 individuals who have made an online purchase. Do you think the necessary assumptions for inference are satisfied? Explain. c) The online shoppers in our sample had an average age of 24.2 years, with a standard deviation of 5.3 years. What’s the p-value for this result? Explain what it means. d) What’s your conclusion?  Example 2 A company with a large fleet of cars hopes to keep gasoline costs down and sets a goal of attaining a fleet average of at least 26 miles per gallon. To see if the goal is being met, they check the gasoline usage for 50 company trips chosen at random, finding a mean of 25.02 mpg and a standard deviation of 4.83 mpg. Is this strong evidence that they have failed to attain their fuel economy goal?  Example 2 a) What are the appropriate hypotheses? b) Are the necessary assumptions to perform inference satisfied? c) Test the hypotheses and find the p-value d) Test the hypotheses using rejection regions and confidence intervals at 1% level of significance.  Example 3 A company that produces cell phones claims its standard phone battery lasts, on average, at least 35 hours. A consumer advocacy group is skeptical of the company’s claim. To investigate this claim, the group takes a random sample of 12 batteries and measures how long the batteries last. The results are as follows: 35,34,32,31,34,34,32,33,35,55,32,31. Is the company’s claim legit? a) Is there anything unusual regarding the data? b) Find a 95% confidence interval and state your conclusion based on it. c) Test the appropriate hypotheses at 5% level of significance based on critical regions.  Example 4 Suppose that you want to estimate the average GPA of the students in the Business School. How large a sample should you choose so that your estimate is within 0.3 of the true average GPA of all business students with 95% confidence. A prior survey yielded a standard deviation of the GPA’s of 0.8.  Example 1 A small college has 240 full-time employees that are currently covered under the school’s health care [plan. The average out-of-pocket cost for the employees on the plan is $1880 with a standard deviation of $515. The college is performing an audit of its health care plan and has randomly selected 30 employees to analyze their out-of-pocket costs. a) Calculate the standard deviation of the sample mean b) What is the probability that the sample mean will be more than $1840? c) What is the probability that the sample mean will be between $1900 and $1950?  Example 2 According to a retail organization, men spent an average of $135 on Valentine’s Day gifts. Assume the standard deviation for this population is $40 and that it is normally distributed. A random sample of 16 men who celebrated Valentine’s Day was selected. a) Calculate the standard deviation of the sample mean. b) What is the probability that the total amount spent by the men exceeds $560?  Example 3 In 2012, a large number of foreclosed homes in the Washington DC metro area were sold. In one community, a sample of 30 foreclosed homes sold for an average of $443705 with a standard deviation of $196196. 1) a) What assumptions and conditions must be checked before finding a confidence interval? How would you check them? b) Find a 95% confidence interval for the mean value per home c) Interpret this interval, and explain what 95% confidence means. d) Suppose nationally, the average foreclosed home sold for $350000. Do you think the average sale price in the sampled community differs significantly from the national average? Explain  Example 5 A company is interested in estimating the cost of lunch in their cafeteria. After surveying employees, the staff calculated that a 95% CI for the mean amount of money spent for lunch over a period of six month is ($780, $920). Now the organization is trying to write its report and considering the following interpretations. Comment on each. a) 95% of all employees pay between $780 and $920 for lunch. b) 95% of the sampled employees pay between $780 and $920 for lunch. c) We are 95% sure that employees in this sample averaged between $780 and $920 for lunch. d) 95% of all samples of employees will have average lunch cost between $780 and $920. e) We are 95% sure that the average amount all employees pay for lunch is between $780 and $920.  Example 6 A company that produces cell phones claims its standard phone battery lasts longer on average than other batteries in the market. To support this claim , the company publishes an ad reporting the results of a recent experiment showing that under normal usage, their batteries last at least 35 hours. To investigate this claim, a consumer advocacy group asked the company for the raw data. The data are as follows: 35, 34, 32, 31, 34, 34, 32, 33, 35, 55, 32, 31 Find a 95% confidence interval and state your conclusion. Explain how you dealt with the outlier, and why