218 MATH SI Session (Spring 2008) SI Leader: Mario Week 11 Worksheet Wednesday (3/26)
Professor: Zygouras/Dumett www.usc.edu/si
1. Raz and Zar are both running for the position of President of Stuffed Animal Kingdom. To win, a candidate must receive more than 50% of the votes. In a pre-election survey of 200 citizens of Stuffed Animal Kingdom. 120 supported Raz while 80 supported Zar. a. Can you conclude with 98% confidence that Raz will be the winner?
b. How many more samples would you have to take if you would like the width of 98% confidence interval to be 10%?
c. Repeat part "b" if you did not know p-hat from the preliminary sample.
2. At Trojan Lake, the new fishing season has arrived. The weights of fish in this lake are normally distributed with a mean of 3 pounds and a standard deviation of 0.75 pounds. a. A sample of 16 fish is caught. Find the probability that the average weight of a fish is at most 2.9 lbs.
b. From a sample of 16 different fish, find the probability that this batch of fish will weight more than 50 lbs.
3. Zygouras Muffin Factory produces a variety of muffins for the staff and students of USC. 25% of the muffins produced are chocolate chip, 15% are oatmeal, 20% are blueberry, and 40% are plain. a. The math department secretary orders 500 muffins (for Math 218 students) without specifying what type she wants. If the muffins are packed at random, what is the probability that there will be exactly 100 blueberry muffins?
b. What is the probability that there will be anywhere between 98 and 102 blueberry muffins?
c. What is the probability that at least 26% of the muffins that arrive are chocolate chip?
d. What is the probability that at most 41% of the muffins are plain?
4. Zygouras Accounting Company has 3% of its employees who have attended graduate school. In Department A of the company, there are 500 employees. a. Find the probability that the proportion of employees in this department who have attended graduate school is less than 4%.
b. In Department B, there are a total of 250 employees. Find the probability that between 4 and 8 (all inclusive) employees from this department have attended graduate school.
c. In Department B, find the probability that exactly 20 of them have attended graduate school.
5. A random sample of 100 Kit Kat Bars is taken and weighted. The sample mean is 41.6 grams and the population standard deviation is 1.5 grams. a. Construct the 95% confidence interval for .
b. Construct the 99% confidence interval for .
6. The student attendance at USC football games follows a normal distribution. In a sample of 9 games, the attendance rates are: 8,000 7,600 8,400 6,120 9,200 7,800 9,400 6,200 8,200 a. Calculate the point estimator for , the population mean.
b. Assume that the standard deviation is 790. Calculate the 95% confidence interval for .
7. Zygouras Motor Tire Inc. uses a tire mold which occasionally, and randomly, produces out-of-round tires. In fact 20% of the tires which it produces are out-of-round. Suppose Slick takes a random sample of 10 tires produced by this tire mold. a. Find the expected number of out-of-round tires among these 10.
b. Find the probability that exactly two of these 10 tires are out-of-round.
c. Find the probability that two or more of these 10 tires are out-of-round.
d. Suppose now that the Fraud Motor Company buys 120 of the tires produced by this tire mold. Find the probability that between 20 and 30 (inclusive) of these tires are out-of-round.
8. Halloween is "coming up" and you need to buy candy for Trick-or-Treaters. Suppose you take a sample of 30 of your neighbors and ask them how many pieces of candy they each gave out last year. The sample mean turns out to be 86 pieces of candy. One of your neighbors, a psychic, tells you that the population standard deviation must be 15. a. Construct a 90% confidence interval for the population mean.
b. Construct a 99% confidence interval for the population mean.
c. How many samples are needed to estimate the 99% confidence interval to be within +/-4?
9. In addition to the number of candies you need to purchase, you would like to find out a little bit about the taste preferences of the Trick-or-Treaters. You sample 100 potential Trick-or-Treaters and ask them whether they prefer chocolate bars, lollipops or gymmy candies. Suppose that 55 Trick-or-Treaters prefer chocolate bars, 24 prefer lollipops, and 21 prefer gummy candies. a. Construct a 95% confidence interval for the proportion of Trick-or-Treaters who prefer chocolate bars.
b. Construct a 98% confidence interval for the proportion of Trick-or-Treaters who prefer gummy candies.
c. How many additional samples are needed to estimate the 98% confidence interval for those who prefer gummy candies so that the width is less than 0.08?
10. Suppose that in a sample of 75 professional baseball players, 24 are lefthanded. a. Find the 90% confidence interval for proportion.
b. Based on the preliminary sample, determine how many more professional baseball players are needed to estimate the proportion to within 0.05.
11. Let X equal the traveling time from home to work for a randomly chosen office worker in Los Angeles. Assume that X has a normal distribution. A sample of 6 observations from this population gave the following times (in minutes): 35 20 25 15 22 21
a. Find point estimates for the population mean and population variance 2.
b. Find a point estimate for the standard deviation of the sample mean X.
c. Find a 95% confidence interval for .
d. Suppose that you are told that = 8. Using this additional information find a 95% confidence interval for .
e. How large should the sample be in order to estimate to within 4 with 95% confidence? (Again, assume that you know that = 8.)
12. A random sample of 100 Big Macs is taken. a. Determine the 95% confidence interval for if the sample mean was 41.6 grams and the population standard deviation is 1.5g.
b. How large should the sample size be to estimate the mean weight of Big Macs to within 0.2g with a 95% confidence interval?
c. How would your answer to part (a) change if you did not know that the population standard deviation is 1.5g and instead found out that the sample standard deviation is 1.5g?
13. Let Y be the time it takes a randomly chosen Math 218 students to complete his or her homework. Assume that Y has a normal distribution. A sample of 6 observations from this population gave the following times (in minutes): 25 21 28 11 29 21
a. Find the mean of the sample.
b. Suppose that you are told that the standard deviation is 6.25; find a 95% confidence interval for the population mean .
c. Find the sample standard deviation.
d. Find a 92% confidence interval for the mean, using the sample standard deviation.
14. The joint distribution of random variables X and Y is given below:
0 0.36 0.18 0.06 X 1 0.18 0.09 0.03 2 0.06 0.03 0.01 marginal
Y
4 5 6 marginal
Assume that E(X) = 4.5 and E(Y) = 0.5. a. Find P(X>5, Y>0).
b. Prove that X and Y are independent.
c. Calculate the covariance of X and Y.
d. Calculate the correlation of X and Y.
15. A survey of Marshall Business School undergraduate students found that 60% of Marshall students live on-campus and 40% live off-campus. Suppose 30 students are chosen at random. a. What is the probability that exactly 10 of them live off-campus?
b. What is the probability that more than 11 of them live off-campus?
c. What is the probability that at most 67% of them live on-campus?
16. Suppose that in a sample of 120 American Idol contestants, 52 are good singers. Find the 91.25% confidence interval for the population proportion.
17. A conference organizer receives registrations with fees by phone, by fax, or via the internet. The registration fees are classified as small (under $200), medium ($200 to $399.99), and large ($400 and up). A study of registration fees yields the following table:
Small 10 40 30 Medium 20 40 40 Large 15 20 5
Phone Fax Internet
a. Find the probability that a randomly selected order was small and received by fax.
b. Find the probability that a randomly selected order was received by fax.
c. Find the probability that a randomly selected order was small or received by fax.
d. Given that a randomly selected order is received by fax, what is the probability that the order is small?
e. Given that a randomly selected order is small, what is the probability that the order is received by fax?
f. Are the events "small" and "received by fax" independent events? Explain.
18. The U.S. justice system follows the principle of "innocence until proven guilty." What are the null and alternative hypotheses of the jury when a defendant is on trial? a. What are the null and alternative hypotheses of the jury when a defendant is on trial? Ho: Ha: b. What are the errors in this example? Type (I) error ( ) : Type (II) error ( ) :
c. Which error is the justice system designed to minimize?
19. The Zygouras Cheese Company sells packages of cheese. These packages are stamped with an expiration date; those packages not sold by the expiration date are discarded. The company claims that on average, the cheese can be kept at least 14 days past the expiration date before going bad. The government believes that cheese cannot be kept 14 days past its expiration. To test this claim, government inspectors decided to keep 30 packages and how record long past its expiration date it takes each package to go bad. The inspectors found that the average time it took the sample of cheeses to go bad was 13 days. Assume that the standard deviation is 2.1 days. a. Formulate the null and alternative hypotheses for this test. Ho: Ha:
b. Choose a test statistic for the test, and evaluate it numerically.
c. Give the rejection rule at the 5% level of significance.
d. What is your conclusion? (Do you reject or not reject Ho?)
20. A quality control manager at a cereal company is concerned about the boxes of cereal being properly labeled. The boxes have been labeled as containing 16 ounces. Assume that the distribution of amounts is normal and that the population standard deviation is 1.0 ounce. The manager is interested in any under-filling on average of the boxes. a. State the appropriate null and alternative hypotheses.
b. Choose a test statistic and rejection rule to test the null hypothesis at the 6% significance level using the sample of 10 boxes.
c. The manager samples 10 boxes, measures the amounts, and find a mean of 15.25 ounces. Should the null hypothesis be rejected? Explain your reasoning.
d. Find the p-value.
e. For which values of the level of significance should the null hypothesis be rejected? Circle all that apply. 0.005 0.01 0.025 0.05 0.075 0.10
f. What conclusion would be a Type I error?
g. Use the following data for the amount of cereal in a box to find the sample mean for the amount of cereal in a box for this company: 13 15 15 16
h. Find the sample standard deviation for the amount of cereal in a box from the sample data in part "g."
21. Dumett Medicine Company has developed a new sleeping pill and has decided to test it on a group of 50 volunteers. It turns out that 7 of them developed a sleeping disorder after taking the pill. a. Next year the pill is going to appear on the market. Find the 90% confidence interval for the proportion of users who develop a sleeping disorder after taking the pill.
b. A scientist analyzes this data. He wants to conduct his own experiment to estimate this proportion to within 0.02. Using the previous data, how many volunteers does the scientist need (with 90% confidence)?
22. After a Pepsi taste test, you believe that you can tell the difference between Pepsi and Coke at any time. However, your friend challenges your ability to distinguish between Pepsi and Coke. He poured Pepsi into 2 glasses and Coke into 2 glasses. Both of you agree that in order for you to prove your ability to distinguish the tastes; you have to tell which 2 glasses contain Pepsi. a. What are the null and alternative hypotheses in this case?
b. What are the rejection criteria? Name two.
c. Describe what the error of Type I would mean in this case. How about Type II?
23. USC's student body is made up of 52% undergraduate and 48% graduate students. A survey is randomly given to 100 USC students. a. What is the probability that exactly 50 of the selected students are undergraduates?
b. What is the probability that at most 51 of the selected students are graduate students?
c. What is the probability that the number of undergraduate students selected us between 49 and 53?
24. In the past, a light bulb had an average lifetime of 100 hours. Because of new technology, a company is able to produce these light bulbs at a cheaper and more efficient rate. To test whether the new technology has increased the average life, data are collected on 49 light bulbs. Assume the life of a light bulb has a normal distribution. a. Formulate the appropriate null and alternative hypothesis.
b. In the sample of 49 light bulbs, there is a mean of 100.8 hours and a sample standard deviation of 4.85 hours. Determine whether you should reject the null hypothesis at the 5% level of significance. Calculate the test statistic and the rejection region.
c. Find the p-value for this test. Does it agree with your answer in part "b"?
d. At which of the following significance levels should you reject Ho? Circle all that apply. 0.5% 1% 2% 5% 10%
25. Zygouras Horse Company has maintained a record of overpaying bills for horse food by a mean amount of less than or equal to $5. A financial inspector believes that the mean overpayment for a normal population of these bills may be larger than $5. To test his claim, the inspector takes a random sample of 9 bills and finds the sample mean of overpayment to be $7 with a standard deviation ( ) of $3. a. State the null and alternative hypotheses.
b. Find the test statistic for this test.
c. If the critical value was found to be 2.3, do we reject the null hypothesis?
d. Find the p-value.
e. Find the probability of Type I error if the population mean overpayment is equal to $5.
26. You survey 100 students to see which of the premium pay television networks shows the best movies. The networks to pick from were HBO, Showtime and Starz! Suppose that 55 of them like HBO the best, 24 prefer Showtime and 21 prefer Starz! a. Find the 93.5% confidence interval for the proportion of students who prefer HBO's movies.
b. Find the 98% confidence interval for the proportion of students who prefer Showtime's movies.
c. How many additional samples are needed to estimate the 98% confidence interval for those who prefer Showtime so that the width is less than 0.08?
27. A researcher for Red Cars of America wants to use a random sample of 426 tickets to see if there is strong evidence to support the idea that an unusually high percentage of tickets go to drivers of red cars. Approximately 11% of the cars on California's roads are red in color. a. Formulate the null and the alternative hypotheses. Ho: Ha: b. Choose an appropriate test statistic and rejection rule to test your null hypothesis against your alternative hypothesis at the 5% level of significance.
c. During a recent 24 hour period, on a particular stretch of highway, the Highway Patrol issued 426 tickets, of which 63 went to drivers of red cars. Determine whether or not the null hypothesis should be rejected at the 5% level of significance.
d. True or False: There is not strong evidence that an unusually high percentage of tickets go to red cars. Red cars apparently receive an unusually high percentage of tickets. Drivers of red cars are treated unfairly by the police.
28. A consumer protection agency took a random sample of a product whose package was marked as weighing 1 pound. The 16 packages in the random sample had a mean weight of 1.10 pounds and a sample standard deviation of 0.36 pound. (Assume normality.) Find a 90% confidence interval for the mean weight of the 1 pound packages.
29. The Zygouras Food Company claims that its cereal boxes contain, on average, 453 grams of cereal. We suspect that the cereal boxes contain, on average, less than claimed. You decide to test the claim by inspecting 6 randomly selected boxes and get the following weights: 454 447 452 446 450 445
a. Formulate the null and the alternative hypotheses. Ho: Ha:
b. Which statistic should be used to test the hypotheses? Evaluate it numerically.
c. Formulate the rejection rule at 5% significance level. Using your computation in "b," decide whether the null hypotheses should be rejected.
d. Calculate the p-value.
e. At which of the following significance level(s) should the null hypothesis be rejected? 0.005 0.01 0.025 0.05 0.1
30. A propane company would like to estimate the mean amount of propane that is stored in the large tanks. The standard deviation of the amount of propane in all the tanks is 50 gallons. How large should the sample be in order to estimate the mean amount of propane in the tanks to be within 10 gallons with 95% confidence?
31. A mail-order business prides itself in its ability to fill customers' orders in six calendar days, on average, the actual number of days being normally distributed. Recently doubts were raised about this claim by some angry customers, since their orders took more than six days to be filled. So the manager took a sample of 25 customers, found that the average number of days was 6.65, with a sample standard deviation of 1.5 days. a. Formulate the null and alternative hypotheses concerning the average number of days required.
b. Is there enough evidence that the angry customers were right? Perform the test at the 5% significance level.
c. If the mail-order business claim were true, the proportion of customers with more than six days delays should be 1/2; but the manager found that it took longer than six days for 18 of the 25 sampled customers. Use the data to check if there is a significantly higher proportion than expected. Use = 5%.
32. An engineer is inspecting steel bar rods in a factory. From a sample of 61 bars, he finds the mean length of them to be 38.9 feet with sample standard deviation of 1 yard. The company owner would like to know how many more samples will be needed to estimate the mean length of the steel bars to be within 0.5 feet with 95% confidence level.
33. At USC, the average height of male students is 70 inches (5' 10"). However, professor Zygouras challenges this number. He believes that the males in his class are over 6 feet tall and that USC's average height number is wrong. He also believes that the average height us greater than 70 inches and will try to prove this by collecting his own data. What are the null and alternative hypotheses in this situation?