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 3.25 Standard Error 0.197 Median 3 Mode 3 Standard Deviation 1.180 Sample Variance 1.393 Kurtosis 0.199 Skewness 0.697 Range 5 Minimum 1 Maximum 6 Sum Count 117 36 Explain and show your working out for all answer. a) Does the above table indicate that Plunket should be concerned that women are not staying long enough in hospital? b) 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. c) 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? d) 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. e) 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?