Problem 1: Consider the data in HospitalStaff.sav to be a random sample for all hospitals in Australia. Use the information in the dataset HospitalStaff.sav to answer the following questions regarding cost of nursing staff (TotalNurseCost) for all Australian hospitals with Psychiatric wards. Note, for this question you will use the data only for the Hospitals with Psychiatric wards.
You should use SPSS to calculate the sample statistics you will need to do this question, but for the confidence interval in part (a) and test statistic in part (d) you are required to do the rest of the calculations by hand, using a calculator.
(a) Find a 99% confidence interval for the population mean cost of nursing staff for all Australian hospitals with Psychiatric wards (show all working). You should use data only from hospitals with Psychiatric wards. [You may use sample statistics from SPSS output without pasting it into your answer.]
(b) Check the appropriate conditions and assumptions needed for the validity of the confidence interval or hypothesis test for the population mean cost of nursing staff for all Australian hospitals with Psychiatric wards (include an appropriate graph to support your answer).
(c) A researcher suspects that the mean cost of nursing staff for all Australian hospitals with Psychiatric wards is smaller than $100 million. State appropriate hypotheses (define any symbols used) to perform a hypothesis test to see if there is evidence to support the suspicion, based on the available data (regardless of whether the conditions in part (b) are satisfied or not).
(d) Calculate the value of the appropriate test statistic for testing the hypotheses in part (c).
(e) Based on the test statistic calculated in part (d) and using the appropriate statistical table provided in the StudyDesk, find the P-value of the test, and write a meaningful conclusion at the 1% level of significance.
(f) Now, check your answers for parts (d) and (e) by finding the value of the test statistic and the P-value using SPSS. Include SPSS output in your answer and comment on the comparison with the hand calculated values. Explain any differences.
Problem 2: Not all hospitals can afford to have an Obstetrics wards due to cost consideration. It is known that in City and Regional areas, 20% of hospitals have an Obstetrics wards. Among the health professionals, there is serious concern that the proportion of hospitals with Obstetrics wards is much lower in the rural area than the City or Regional areas. Assume that the data is a random sample of all hospitals in Australia.
Use the information in the dataset HospitalStaff.sav to answer the following questions: You should use SPSS to calculate any sample statistics you will need to do this question, but for parts (d)-(g) you are required to do the rest of the calculations by hand, using a calculator and statistical tables.
(a) What is the variable of interest here?
(b) State the appropriate hypotheses (define any symbols used) to test the concern of the health professionals that the proportion of hospitals with Obstetrics wards is lower than 20% in the Rural areas of Australia. Note, you will use data only from rural hospitals for this question.
(c) Check the conditions and assumptions for the hypotheses to be tested in part (b).
(d) Calculate the value of the appropriate test statistic for testing the hypotheses in part (b).
(e) Using the appropriate statistical table provided in the StudyDesk, find the P-value for the test, and write a meaningful conclusion at the 5% level of significance in the context of this study.
(f) If a health professional wants to be 95% confident that the margin of error of the estimate of the true proportion of rural hospitals with obstetrics wards in Australia is within 0.06, what minimum sample size is required? For calculations, use an estimated proportion from the given data.
(g) If the health professional decides to use a conservative method (approach), what will be the minimum sample size to keep the same level of confidence and margin of error as in part (f)? What is the impact of this decision? (Include evidence to support your answer).
Problem 3: A brain specialist performed an experiment to see if sensory deprivation over an extended period of time reduces the alpha-wave frequencies produced by the brain. To determine this, 10 subjects, inmates of a Correctional Centre, were randomly selected. First the members of the group were allowed to remain in their cells. Seven days later, alpha-wave frequencies were measured in Hertz for all subjects (non-confined). After that the same group of subjects was placed in solitary confinement for seven days and their alpha-wave frequencies were measured again in Hertz. The data from both confined and non-confined groups of the experiment are recorded in the following table.
|
Subject #
|
Confined
|
Non-confined
|
|
1
|
7
|
8
|
|
2
|
10
|
9
|
|
3
|
9
|
10
|
|
4
|
9
|
10
|
|
5
|
10
|
11
|
|
6
|
9
|
11
|
|
7
|
11
|
11
|
|
8
|
12
|
11
|
|
9
|
11
|
12
|
|
10
|
12
|
14
|
(a) Using a parametric test determine if prolonged solitary confinement reduces the alpha-wave frequencies of the brain. State appropriate hypotheses (define any symbols used) to perform this test.
(b) State (but do not check) the assumptions for carrying out the test in (a). Describe the assumptions in the context of this question.
(c) Without using SPSS, calculate the value of a suitable test statistic for the test in part (a). You can use SPSS for calculating appropriate sample statistics, but the value of the appropriate test statistic must be calculated showing working.
(d) Without using SPSS, calculate the P-value of the test in (c).
(e) Interpret the P-value and describe the outcome of the test in the context of this question.
(f) Use a non-parametric test to investigate if the solitary confinement reduces the alpha-wave frequencies of the brain. In performing this test, state appropriate hypothesis (define any symbols used). Calculate the value of the test statistic, P-value of this test and write a meaningful conclusion.
(g) Comment on any differences found between the outcomes of the two tests and why this may be the case.
Problem 4: A researcher suspects that the 2012 total nurses cost in million dollar (TotalNurseCost) in all Australian hospitals varied among the three regions - city, regional and rural. Use the information in the dataset HospitalStaff.sav to answer the following questions.
You should use SPSS to produce the graph and calculate the sample statistics, but for part (e) you are required to do all other calculations by hand, using a calculator.
(a) State appropriate hypotheses (defining all symbols) to answer the question: 'Is the mean 2012 total nurses cost in million dollar (TotalNurseCost) less in hospitals without obstetrics wards than those with obstetrics wards?'
(b) State (but do not check) the assumptions required for the validity of the test. Describe the assumptions in the context of the study.
(c) Without using SPSS, calculate the value of the appropriate test statistic for testing the hypotheses in part (c). [You can use appropriate sample statistics from SPSS output for calculations.]
(d) Using the appropriate statistical table provided in the StudyDesk, find the P- value of the test, and describe the outcome of the test in the context of the study.
(e) Now use SPSS to check your results for the above hypothesis test. Copy and paste the relevant output from SPSS for this test into your assignment.
(f) Briefly comment on how the test statistic and P-value from SPSS output are similar to or differ from those of your by hand calculations.
Problem 5: The waiting time in the emergency department in a large hospital is a concern for the outdoor patients. Based on the historical records of the hospital it is found that the mean and standard deviation of waiting time of patients in the emergency department are 40 minutes and 6 minutes respectively. Assume that the distribution of waiting time follows a normal model. For the waiting time of a random sample 25 patients from the population of patients in the emergency department of the hospital, answer the following questions.
(a) What is the variable of interest in the question?
(b) Find the probability that one randomly selected patient will wait less than 36 minutes.
(c) What type of distribution is the sampling distribution of the waiting time of the sample means of patients? Specify the parameters of this sampling distribution for a random sample of 25 patients from the population of all patients in the emergency department in the hospital.
(d) Find the probability that the sample mean of a random sample of 25 patients will be less than 36 minutes.
(e) Explain the reason for the difference in your answers in parts (b) and (d).
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