Mathematics
22)
Market observers are quite uncertain whether the stock market has bottomed out from the economic meltdown that began in 2008. In an interview on March 8, 2009, CNBC interviewed two prominent economists who offered differing views on whether the U.S. economy was getting stronger or weaker. An investor not wanting to miss out on possible investment opportunities considers investing $20,000 in the stock market. He believes that the probability is 0.30 that the market will improve, 0.37 that it will stay the same, and 0.33 that it will deteriorate. Further, if the economy improves, he expects his investment to grow to $28,000, but it can also go down to $17,000 if the economy deteriorates. If the economy stays the same, his investment will stay at $20,000. 
a.  What is the expected value of his investment? 
Expected value  $ 
b.  What should the investor do if he is risk neutral? 
Investor invest the $20,000. 
c.  Is the decision clearcut if he is risk averse?  

23)
A car manufacturer is concerned about poor customer satisfaction at one of its dealerships. The management decides to evaluate the satisfaction surveys of its next 66 customers. The dealer will be fined if the number of customers who report favorably is between 26 and 33. The dealership will be dissolved if fewer than 26 report favorably. It is known that 62% of the dealer’s customers report favorably on satisfaction surveys. Use Table 1. 
a.  What is the probability that the dealer will be fined? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.) 
Probability 
b.  What is the probability that the dealership will be dissolved? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.) 
Probability 
24)
The historical returns on a balanced portfolio have had an average return of 10% and a standard deviation of 14%. Assume that returns on this portfolio follow a normal distribution. Use the empirical rule for normal distributions to answer the following questions. 
a.  What percentage of returns were greater than 38%? (Round your answer to 1 decimal place.) 
Percentage of returns  % 
b.  What percentage of returns were below −18%? (Round your answer to 1 decimal place.) 
Percentage of returns  % 
25)
A machine that is programmed to package 5.35 pounds of cereal is being tested for its accuracy. In a sample of 16 cereal boxes, the sample mean filling weight is calculated as 5.35 pounds. It can be assumed that filling weights are normally distributed with a population standard deviation of 0.04 pound. Use Table 1. 
a1.  Identify the relevant parameter of interest for these quantitative data.  

a2.  Compute the point estimate as well as the margin of error with 95% confidence. (Round intermediate calculations to 4 decimal places. Round “z” value and final answers to 2 decimal places.) 
Point estimate  
Margin of error  
b1.  Calculate the 95% confidence interval. (Use rounded margin of error. Round your answers to 2 decimal places.) 
Confidence interval  to 
b2.  Can we conclude that the packaging machine is operating improperly?  

c.  How large a sample must we take if we want the margin of error to be at most 0.01 pound with 95% confidence? (Round intermediate calculations to 4 decimal places. Round “z” value to 2 decimal places and round up your final answer to the next whole number.) 
Sample size 
26)
Consider the following hypotheses: 
H0: μ ≥ 100 
HA: μ < 100 
The population is normally distributed. A sample produces the following observations: 
88  77  100  83  102  96 
Use the critical value approach to conduct the test at a 5% level of significance. Use Table 2. 
a.  Find the mean and the standard deviation. (Round intermediate calculations to 4 decimal places. Round your answers to 2 decimal places.) 
Mean  
Standard deviation  
b.  Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.) 
Test statistic 
c.  Calculate the critical value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 3 decimal places.) 
Critical value 
d.  What is the conclusion?  
