problem 1: For each of the statements below, prepare down whether it is true or false.
i) We are conducting a 2 tailed hypothesis test at a 1% significance level to test that the mean of a population is 500 against the alternative hypothesis that the mean is not the same. The correct null and alternative hypotheses are:
H_{o} : x‾ = 500 H_{A} : x‾ ≠ 500.
ii) An experiment involves testing whether international soccer players are different in height from the rest of the population. A type II error in a two tailed-test for this problem would conclude that soccer players are the same height as the rest of the population when they are not.
iii) When using the p-value to test hypotheses, the decision rule depends only on the level of significance of the test.
iv) The standard error of the mean will increase when the sample size is increased.
v) As long as a sample is large enough, the shape of a sampling distribution of the sample proportion is approximately normal.
vi) We are using a t-test to test a hypothesis about a population mean. The degrees of freedom of the critical value depend on the sample size.
vii) The estimate of the standard error of an unknown population proportion is √(pq)/n
viii) The data collected by means of a survey are described as primary data.
ix) Simple random sampling is the method for obtaining the sample used to conduct the Australian Census.
x) The difference between a sample mean and the population mean arises from sampling errors only.
problem 2:
a) A beverage company uses a machine to fill its 500 ml cartons. The machine is calibrated to fill the cartons with an average of 505 ml and a standard deviation of 5 ml.
i) A carton is randomly selected from a day’s production, what is the probability that it has been under filled?
ii) A random sample of 50 bottles is taken from a day’s production. What is the probability that the average fill volume is less than 500 ml?
iii) Based on your answers to part (i) and part (ii), what comment could be made about the company’s calibration standard?
problem 3:
Many consumers use credit cards for their convenience. A sample of 100 credit card holders finds that only 35% of them pay the card off each month.
a) Construct a 95% confidence interval estimate of the true proportion of credit card holders who pay off their credit cards each month.
b) Use the appropriate Excel workbook from Estimators.xls to construct a 99% confidence interval estimate of the proportion of credit holders who pay off their credit cards each month. Include the resulting Excel output with your answer. Highlight the upper and lower limits of the confidence interval in this output.
c) Which of the intervals in parts a. and b. is wider? prepare a sentence to describe why this is.
problem 4
a) A major bank is evaluating its recently revamped online banking system. Management will conclude there to be a problem with the system if less than 95% of users are satisfied with the service. 3000 customers were contacted but only 1568 agreed to be surveyed. Of these, 1452 were satisfied with the service they received. Do the sample data provide sufficient evidence to conclude that management has a problem with the online banking system? Use α = 0.05
State clearly the null and alternative hypotheses, the test statistic, test result, decision rule and conclusion in terms of the original problem.
b) What is the approximate p-value of the test in part a.? No working required.
c) How could this p-value be used to formally test the same hypotheses described in part a.? Just include the decision rule and the conclusion and a justification for the conclusion.
d) Suppose we wish to estimate the percentage of customers who are satisfied with the service to within 1%, within 95% confidence. What sample size would be required? Use the sample proportion from part a. when calculating the appropriate sample size.
problem 5:
A simple random sample of the birth weights of 100 babies in a major teaching hospital gave a mean weight of 3567 gm with a standard deviation of 492 gm.
a) Construct a 99% confidence interval estimate of the population mean birth weight of babies born in this hospital.
b) Use the appropriate Excel workbook from Estimators.xls to construct the same interval as was required in part a. Include the resulting Excel output. Highlight the upper and lower limits of the confidence interval in this output.
Note: This should provide a check of your calculations in part a. If there are big differences you should check both your manual calculations and your Excel calculations to determine where the error has occurred.
problem 6:
An importer of women’s tracksuits needs to check that the average height of adult female basketball players is still 170cm. The company measures the heights of a random sample of 60 basketball players and finds the mean to be 175.4cm. Assume that the historic value for the standard deviation of 85.2 cm is unchanged.
a) Using a 0.05 level of significance, is there evidence that the heights of adult female basket ballers has changed?
State clearly the null and alternative hypotheses, the test statistic, test result, decision rule and conclusion in terms of the original problem.
b) Use the appropriate Excel macro in Test Statistics.xls to again perform the hypothesis test in part a. Present the output generated by Excel as your answer here.
c) describe how this output verifies your conclusion in part a. by reference to either z-stat and z-critical or the p-value of the test.