Supplementary exercise for students with excessive absences. You are to complete all the exercises below and submit them on the last day of class, December 13, 2010. Each solution is to be clearly numbered. Show all work to receive full credit.
1. An instant lottery ticket costs $1. In a total of 10,000 tickets for this lottery, 1000 tickets contain a prize of $3 each, 100 tickets have a prize of $5 each, five tickets have a prize of $100 each, and one ticket has a prize of $1000. Let x be the random variable that denotes the net amount a player wins by playing this lottery.
a. Write the probability distribution of x.
b. Determine the mean and standard deviation of x.
c. How will you interpret the value of the mean of x?
2. Sixteen percent of adults contribute to charitable agencies on a regular basis. Using the binomial formula, find the probability that in a random sample of 12 adults, exactly five contribute to charitable agencies on a regular basis.
3. Sixty-four percent of all airplanes arriving at an airport are late. Using the binomial formula, find the probability that in a random sample of seven airplanes, exactly four will arrive late.
4. Players of a casino game pick a number from 1 to 36. A roulette wheel is then spun, and players win if their number comes up, and lose if it does not. Players can choose only one number per spin. Assume that a player plays the game 360 times. Using the normal approximation to the binomial distribution, what is the probability that the player wins exactly nine times?
5. The mean annual water bill for all households in a community is $480. Assume that the annual water bills for all households in the community follow a normal distribution with mean $480 and standard deviation $100. The city planner is interested in houses that spend either less than $200 per year or more than $600 per year. What is the probability that a randomly-selected household from the community falls into one of these categories?
6. Course times on a particular 10-kilometer run approximate a normal distribution with a mean of 35 minutes, 24 seconds and a standard deviation of 10 minutes, 42 seconds. Runners whose times are 31 minutes or faster qualify for the state competition. What is the probability that a randomly-selected runner in the race will qualify for the state competition?
7. A man walking along the railroad tracks near his town has gotten halfway across a very high bridge when he finds that the train is approaching from the east. His only chance for survival is to run toward the west end of the bridge and leap to the side of the tracks before the train gets there. He can’t tell how fast the train is going, but he knows that the average speed of trains through the area is 60 miles per hour. If the train is going 65 miles per hour or slower, the man will make it to safety. Assuming that the train speed follows a normal distribution with mean 60 and standard deviation 10, what is the probability that the man will make it to safety?
8. The mean number of points per game for all 250 players in a certain league is 8.4, with a standard deviation of 2.3 points. Let be the mean of a random sample of players chosen from the league. Find the mean and standard deviation of for a sample size of:
a. 4 c. 13
b. 12 d. 20
9. The time required for all cross-country skiers to complete a race has a normal distribution with a mean of 46 minutes and a standard deviation of 6 minutes. You choose a random sample of eight skiers, and the mean of the sample is denoted . What are the mean and standard deviation of ?
10. The mean number of hours per week spent studying for students at a large state university is 26 with a standard deviation of 12. However, the distribution is right-skewed. You decide to take a random sample of students. Calculate the mean and standard deviation of , the sample mean, and describe the shape of the sampling distribution when the sample size is:
11. In a poll of 1500 adults, 28% said that they live a stressful life. Construct a 99% confidence interval for the proportion of all adults who live a stressful life.
12. A researcher wants to make a 95% confidence interval for the population mean. The population standard deviation is 25.50. What sample size is required to obtain an estimate that is within 5.50 of the population mean?
13. A researcher wants to make a 99% confidence interval for the population mean. The population standard deviation is 11. What sample size is required to obtain an estimate that is within 3.50 of the population mean?
14. A scientist is studying the paramecium, a one-celled organism, under a microscope. There are 1500 paramecia in the slide he is studying, and the standard deviation of their lengths is .12 mm. He views a sample of 30 paramecia, and finds that the mean length of these 30 specimens is .25 mm.
a. What is the point estimate of the mean length of the entire paramecium population in the slide?
b. What is the margin of error for this estimate?
c. Construct a 95% confidence interval for the mean length of the population.
15. A particular country contains 4000 mountain peaks. A sample of 36 of the country’s peaks is studied, and the mean and standard deviation of the heights of these peaks are calculated. The sample mean is found to be 6240 ft. The boundaries of a 99% confidence interval for the mean height of all 4000 peaks are 6214.2 ft and 6265.8 ft. What is the standard deviation of the sample?
16. A researcher claims that the starting mean salary for new college graduates is $26,700. A sample of 100 recent college graduates showed that their mean starting salary is $28,200 with a standard deviation of $2700. Test at the 5% significance level if the mean starting salary of new college graduates is different from $26,700.
17. A manufacturer claims that its soap bar has a mean net weight of 4.50 ounces. A consumer agency wants to test this claim. A sample of 80 soap bars of this brand gave a mean net weight of 4.46 ounces with a standard deviation of .06 ounces. Test at the 1% significance level if the mean net weight of these soap bars is less than 4.5 ounces.
|Due By (Pacific Time)||10/28/2013 12:00 am|
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