Please answer all three questions.

Background-

A health agency is taking a survey regarding the evaluation of all the hospitals in Melbourne to find out the statistical calculation and analyze the attributes of the data. The agency needs a report that can analyze the number of admissions, type of control and type of service and other factors. They focus on the service provided by each hospital whether it is for non-government, non-federal, for profit, federal government and so on.

For analytical purposes a random sample of 60 data is generated out of 300 population data and the statistical summary of the sample data is tabulated in Table 1. 

Your overall task is to investigate the effect of various variables on the number of admissions in Melbourne hospitals using various descriptive analyses which is extremely helpful to formulate a conclusion. 

To assist with your investigation, you are required to answer questions 1 and 2 below.

Question 1-

a) For a series of random samples of 60, are the mean values of these random samples normally distributed? Explain 

b) Calculate the standard error of the mean and explain the meaning of this value. 

c) Determine the 95% confidence interval and explain its meaning in the context of the overall problem.

d) What is the probability that a sample of 60 hospitals selected at random in the Melbourne area will have a mean greater than 7000.00 admissions?

e) If the admissions times were more variable, what effect would this have on the confidence interval?

Question 2-

Assume that the average admission for all hospitals in Melbourne is 7500. Conduct a statistical hypothesis test to determine if the admission of hospitals in Melbourne is significantly different from the average admission 6959.  Mention any assumptions and include relevant hypotheses and report the results and conclusion in the conventional manner. 

a) Write down both the null and alternative hypotheses

b) Carry out the t test and report the p-value, and the test statistic

c) Write an appropriate conclusion in the context of the problem. 

Based on your answers to questions 1 and 2 please write a report of the effect of various variables on the number of admissions in Melbourne hospitals

Question 3-

 Most of the time houses prices depend on the local market conditions. In addition one of the factors is the number of bedrooms (as bedrooms increase prices increases). Recently Come Real Estate Agency has conducted a survey and selected a random sample of 211 for July 2015 sale in Melbourne and the data analyzed is summarized as follows. 

a) Write down the regression equation.

b) State the R-squared value and the standard error and explain what they mean with respect to the data. 

c) Write down the value of the gradient of the regression line and explain what it means for this data.

d) Are the values for the constant and the gradient (slope) significant (i.e. significantly different from zero) in this case? Justify your answer. 

e) Conduct a hypothesis test on the slope coefficient to test whether there is a linear relationship between number of bedrooms and prices of the houses. Include the null and alternative hypotheses; key test results and an appropriate conclusion.

f) Does the linear regression provide a good model? Give statistical reasons based on the scatterplot, p-values, the standard error and coefficient of determination.

g) If you were developing a model to predict the prices of the houses on the number of bedrooms, what other factors would you like to be able to include?