Question 1[20 points, 5 marks each]
a) Each member of a random sample of 20 business economists was asked to predict the rate of inflation for the coming year. Assume that the predictions for the whole population of business economists follow a normal distribution with a standard deviation of 2%.
- i. The probability is 0.01 that the sample standard deviation is larger than what value?
- ii. The probability is 0.025 that the sample standard deviation is smaller than what value?
b) Monthly rates of return on the shares of a particular common stock are independent of one another and normally distributed with a standard deviation of 1.8. A sample of 15 months is taken.
- i. Find the probability that the sample standard deviation is less than 2.6.
- ii. Find the probability that the sample standard deviation is more than 1.2.
Question 2[20 points]
a) [10 points]To calculate a required sample size to estimate a population proportion, given a desired confidence interval and margin of error, the sample proportion is required but often unknown before the sample is collected. How is this predicament resolved?
b) [5 points]How large a sample is needed to estimate the population proportion if ME = 0.09, = 0.05?
c) [5 points]A fast food company wants to determine the average number of times that fast food eaters visit fast food restaurants per week. They have decided that their estimate needs to be accurate to within plus or minus one-tenth of a visit, and they want to be 90% sure of this. Previous research has shown that the standard deviation is 0.7 visits. What is the required sample size?
Question 3 [25 points]
A company selling licenses for new e-commerce computer software advertises that firms using this software obtain, on average during the first year, a yield of 10% on their initial investments. A random sample of 10 of these franchises produced the following yields for the first year of operation:
6.1 9.2 11.5 8.6 12.1 3.9 8.4 10.1 9.4 8.9
a) [4 marks] Calculate the sample mean and sample standard deviation.
b) [4 marks] Assuming that the population distribution is normal, construct a 95% confidence interval for the true standard deviation of the yield on initial investment during the first year.
c) [2 marks] Assuming that the population distribution is normal, formulate a suitable null and alternative hypothesis to test the company’s claim at the 5% level of significance.
d) [2 marks] What is the test’s decision rule?
e) [2 marks] Calculate the test statistic.
f) [1 mark] Based on the test statistic calculated in (d) above, what is your conclusion about the company’s claim?
g) [2 marks] Calculate the p-value for the test statistic computed in (d) above.
h) [1 mark] Using the p-value found in (f) above, what is your conclusion about the company’s claim?
i) [3 marks] Construct a 95% confidence interval for the average yield on initial investment during the first year for all firms using the company’s software.
j) [3 marks] Does the confidence interval in (h) above support your conclusion from the hypothesis test? Explain.
Question 4[16 points, 8 points each]
a) A music industry professional claims that the average amount of money that an average teenager spends per month on music is at least $50. Based upon previous research, the population standard deviation is estimated to be $12.42. The music professional surveys 20 students and finds that the mean spending is $47.77. Is there evidence that the average amount spent by students is less than $50? Using the p-value approach, test at the 10% significance level.
b) The manufacturer of a new chewing gum claims that at least eight out of ten dentists surveyed prefer their type of gum and recommend it for their patients who chew gum. An independent consumer research firm decides to test their claim. The findings in a sample of 400 doctors indicate that 76% do actually prefer the manufacturer’s gum. Is this evidence sufficient to cast doubt on the manufacturer’s claim? Use α = 0.05.