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**Question 1**

The aptitude test scores of applicants to a university graduate program are normally distributed with mean 500 and standard deviation 60.

a) Applicants need a test score higher than 530 to be admitted into the graduate program. What proportion

of applications qualify? [Marks 2]

b) If the university wishes to set the cutoff score for graduate admission so that only the top 10% of applicants qualify for admission, what is the required cutoff score? [Marks 2]

c) What percentage of applicants have test scores within two standard deviations of the mean? [Marks 3]

**Question 2**

Government officials in Canberra have recently expressed concern regarding overruns on military contracts. These

unplanned expenditures have been costing Australians millions of dollars every year. The prime minister impanels a committee of experts to estimate the average amount each contract costs the government over and above the amount agreed upon. The committee has already determined that the standard deviation in overruns is $17.5 million, and that they appear normally distributed.

i. If a sample of 25 contracts is selected, how likely is it the sample will overestimate the population mean by more than $10 million? [Marks 3]

ii. The prime minister will accept an error of $5 million in the estimate of μ. How likely is he to receive an estimate from the committee within the specified range? [Marks 4]

**Question 3**

a) You have just graduated with a post graduate degree in business and have obtained a position with a large manufacturing firm. The director of marketing has asked you to estimate the mean time required to complete a particular unit of the manufacturing process. A sample of 600 units yields a mean of 7.2 days. Since the population standard deviation is unknown, the sample standard deviation of s=1.9 days must be used. Calculate and interpret the 90 percent interval for the mean completion time for the manufacturing process. If this mean time is estimated

to be in excess of 7 days, a new process will be implemented to reduce production costs. [Marks 4]

b) A construction firm was charged with inflating the expense vouchers it files for construction contracts with the federal government. The contract states that a certain type of job should average $1,150. In the interest of time, the directors of only 12 government agencies were called on to enter court testimony regarding the firm’s vouchers. If a mean of $1,275 and a standard deviation of $235 are discovered from testimony, would a 95 percent confidence interval support the firm’s legal case? Assume voucher amounts are normal. [Marks 4]

**Question 4**

a) The mean size of commercial loans by a bank has been $60,000 in the past. A recent change in the bank’s credit policy allows larger amounts to be borrowed under the same terms. The credit manager now wishes to test whether the mean size of commercial loans made since the policy changes is larger than $60,000. The manager wishes to control the α risk at 0.01 when μ=$60,000. A random sample of n=144 loans made since the policy change yielded

the following results =$68,100, s=$45,000.

i. Conduct the test. State the alternatives, the decision rule, the value standardised test statistic and the conclusion. [Marks 4]

ii. Calculate the p-value of the test and conduct the required test. Interpret its meaning here. [Marks 4]

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