Mathematics
1. A local bank has 5276 accounts cross-classified by type of account and average account balance. The summarized results are (in dollars):
ACCOUNT CHECKING SAVINGS NEW MONEY MKT. TOTAL
BALLANCE ACCOUNT ACCOUNTACCOUNTACCOUNT¬¬¬¬¬¬¬__________
< 500 803 21 90 1934
500 – 1000 640 452 112
>1000 51 538 116 1364
TOTAL 2236 5276
a. Complete the missing parts of the contingency table and answer the questions below.
b. What is the probability that an account does not have over $1000 in it and the account is not a money-market account?
c. What is the probability that a new account’s balance is between $500 and $1000?
d. Given that an account is not a savings account, what is the probability that the account has $1000 or less?
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2. A lawyer estimates that 40% of the cases in which she represented the defendant were won. If the lawyer is presently representing ten defendants in different cases, what is the probability that at least five of the cases will be won? What are you assuming here?Hint: binomial distribution
3a. Find the area to the right of z = 1.27
b. Find the area to the left of z = -0.48
c. Find the area between z = -1.04 and z = .75
d. Find z such that the area to its left is 0.2836.
e. Find z such that the area to its right is 0.7878.
4. The estimated miles-per-gallon (on the highway) ratings of a class of trucks are normally distributed with a mean of 12.8 and standard deviation of 3.2. What is the probability that one of these trucks selected at random would be between 13 and 15 miles-per-gallon?
5. The test scores for the MZZZ admission test are normally distributed with a mean of 450 and standard deviation of 100. A certain highly ranked universitywould not accept anyone scoring below 480 in this test. What percentage of the applicants would be acceptable to the University?
6. Electricalc has determined that the assembly time for a particular electrical component is normally distributed with a mean of 20 minutes and a standard deviation of 3 minutes. What is the probability that the sample mean(average)assembly time for 15 employees exceeds 22 minutes?
7. The weights bags of carrots are normally distributed with a mean of 32 ounces and standard deviation of 0.36 ounce. Bags in the upper 4.5% are too heavy and must be repackaged. What is the least a bag of baby carrots must weigh if it has to be repackaged?
8. A sample of 64 customers at Ron and Ted’s Service station showed the total number of gallons of gasoline purchased by the customers to be 870.40 gallons. If the population standard deviation of gallons was known to be 3.0, find an 87.5% confidence interval estimate for the mean number of gallons purchased.
9. A certain company makes light fixtures on assembly line. An efficiency expert wants to determine the mean number of time it takes an employee to assemble the switch on one of these fixtures. A preliminary study used a random sample of 45 observations and found that the sample standard deviation was 78 seconds. How many more observations are necessary for the efficiency expert to be 97% sure that the point estimate of the true mean will be off by at most 15 seconds?
10. A national survey of 1034 parents showed that 848 claimed pizza was their children’s favorite food (USA Today, January 14, 1992).
a. Letp represent the proportion of all parents who claim that pizza is their children’s favorite food. Find a point estimate for p.
b. Find an 85% confidence interval for p. Is the use of the normal approximation to the binomial justified in this problem? Explain.
11. A snack-food company produces a 454 g (gram) bag of pretzels. Although the actual net weights deviate from 454 g and vary from one bag to another, the company insists that the mean weight of the bags be kept at 454 g. Indeed if the mean net weight is less than 454 g, the company will be short-changing its customers; and if the mean net weight exceeds 454 g, the company will be unnecessarily over filling the bags.
As part of its program, the quality assurance department periodically performs a hypothesis test to decide whether the packaging machine is working properly, that is, to decide whether the mean net weight of all bags packaged is 454 g.
a. Determine the null hypothesis for the test.
b. Determine the alternative hypothesis for the test.
c. The bags of pretzels, according to the quality control department, were found to be normally distributed with a mean net weight of454 and standard deviation of 7.8 g. A random sample of 32 bags of pretzels was found to have a mean net weight of 450 g. Do the data provide sufficient evidence to conclude that the packaging machine is not working properly at the 5% level of significance?
d. What is the p-value of the test?
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12. According to Nielson Media Research, in 1998, during the time slot from 8:00 P.M. to 11:00 P.M., the average person watched 7 hours and 43 minutes of TV per week. A random sample of 25 women in the age group 18 – 24 years yielded a sample mean TV watching time of 343.25 hours and standard deviation of 222.20 hours. Do the data provide sufficient evidence to conclude that during the time slot from 8:00 P.M. to 11.00 P.M., women in the age group 18 – 24 years watched less TV on average than people in general? Perform the hypothesis test at the 1% significance level. b. What is the p-value of the test?