1.   A study reported in the Wall Street Journal found that of 899 home-based businesses,

369 were women-owned. This raises the question; do men or women own an equal number of home-based businesses?

a)         In terms of proportions, state the appropriate null and alternative hypotheses and explain in words what they mean.

b)        Define the acceptance and rejection region if a level of significance of 0.05 is selected. You first have to decide if you can use the normal distributions to    approximate the sampling distribution of proportions.

e)         If the p-value were calculated to be 7.9064E-08, would you accept or reject the null hypothesis?

2.   The Plunket Society (looks after babies and mothers at home in first 12 months) is      interested in estimating the average number of maternity days women stay in the local maternity hospital. Assume Plunket believe that the outcome is better for women and their baby if they stay at least 5 days in hospital. A random sample is taken of 36        women who had babies in Auckland Hospital during the past year. The following       table gives the key statistics for this sample

Auckland Hospital

Mean

Standard Error

Median

3.25

0.197

3

Mode                                        3

Sum                                      117

Count                                     36

Does the above table indicate that Plunket should be concerned that women are not staying long enough in hospital?

ii)        Determine the 90% confidence interval for the mean for all women based on  this sample of 36 women and assuming that the population standard deviation is 1.5 days. Does this sample support the belief of the Plunket society that      women are being discharged too early? Explain.

iii)       Use a sketch diagram to show the difference between the normal probability distribution and the student’s t distribution to explain how the above             confidence interval would be expected to change?

iv)       What is the appropriate sample size for the survey if they were happy to accept a margin of error of half a day (0.5) and still be 90% confident of their interval of mean days stayed? Would the sample be larger or smaller if you wanted to  be 95% confident?   Explain.

v)        Use the following table to test at the  α = 10% level of significance,  whether there is a significant difference between the length of stay at Auckland Hospital (A) and Waikato Hospital (W) using either the appropriate t-values or the p- value method. Make sure you clearly state your null and alternate hypotheses and conclusion to your test

t-Test: Two-Sample Assuming Unequal

Variances

alpha = 0.1

vi)       Briefly explain when you could make a Type II error in the above hypothesis   test. What would the implication of this error be for the Plunket Society if they were aiming to show that there was a difference in mean days stayed at the      two maternity hospitals?