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ECOM90009
ASSIGNMENT QUESTIONS
Question 1
Background
The basis or foundation for statistical inference is the concept of a sampling distribution. This is the probability distribution of the values that a sample statistic such as the sample mean could take if we were to look at every possible sample. According to the Central Limit Theorem the Sampling Distribution of the values the sample mean associated with every possible sample will have a Normal distribution if either the population the sample is taken from is Normal or if the sample size is large enough. For this Sampling Distribution the mean will be equal to the mean of the original population and the variance will be equal to the variance of the original population divided by the size of the sample
In this assignment we will consider two different situations where we imagine that samples are taken from a population that does not have a Normal distribution. Instead in this population the possible values are the integer values that represent the number of people who buy our product whenever we approach a group of n = 12 people. For the 13 different possible integer values from 0 to 12 the probabilities are given by the Binomial distribution in which the probability of success is p = 0.25.
We will look at what happens when we generate 5000 samples of size 10 and also what happens when we generate 5000 samples of size 50 using the Random Number Generator in Excel’s Data Analysis. For each possible sample we calculate several sample statistics starting with the sample mean . We then look at the sampling distributions of these sample statistics we obtained from these 5000 randomly selected samples to see whether or not it is consistent with what the theory says it should be.
To generate 5000 possible random samples of size n = 10 where the original population the samples are taken from has a Binomial distribution with n = 12 and p = 0.25 in Excel click
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Data / Data Analysis / Random Number Generation
and then make the entries shown on top of the next page.
The n = 10 values for each of the 5000 possible samples are now stored in cells A2:J5001
The possible values in these samples are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.
Answer the Following Questions
(a) Using the information you have been given find the population mean m and the population variance s2 using the formulae for a Binomial distribution
Use Excel to find the probabilities of the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.
Using these probabilities estimate the population mean m and the population variance s2 and check whether your answers agrees with what your obtained with the formulae.
(5 marks)
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(b) For each of the 5000 random samples of size n = 10 you have generated, using the values for n and p from part (a) of this question find the following sample statistics.
The sample mean
The sample median Md
The Z statistic for the sample mean Z =
The unbiased estimate of the variance for each sample s2.
The t statistic for the sample mean t(n-1) =
The Chi-square statistic =
After you have obtained the 5000 values of each of these statistics find the Histograms and the Descriptive Statistics for the sample means, the sample medians, the Z statistics, the t statistics and the Chi-square statistics.
Question 1 (b) (contd.)
Briefly discuss whether the shapes of the Histograms and the values of the Descriptive Statistics are or are not consistent with what you were told in lectures about the distributions these statistics are supposed to have.
You should also use the appropriate tables to find the critical values for the z, t(n-1) and statistics associated with a probability of 0.05 in both the left hand tail and in the right hand tail. You should then check what percentage of your simulated values lie in these critical regions.
Is there any reason why your results might differ from what the theory says the results should be? (20 marks)
Total marks 25 = 5 + 20
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Question 2
Suppose you decide to generate 5000 random samples from the same population which has a Binomial distribution with n = 12 and p = 0.25. In this question however you must now generate 5000 random samples of size n = 50.
- Before you examine the 5000 possible values briefly discuss whether the Sampling distribution of the possible values that you will now obtain will be the same as or different from what you obtained in Question 1 (b). If you expect the Sampling distribution of the mean to be different state how you expect it to be different. (5 marks)
(b) For each of the 5000 random samples of size n = 50 you have generated, using the values for n and p from Question 1 (a) find the following sample statistics.
The sample mean
The sample median Md
The Z statistic for the sample mean Z =
The unbiased estimate of the variance for each sample s2.
The t statistic for the sample mean t(n-1) =
The Chi-square statistic =
After you have obtained the 5000 values of each of these statistics find the Histograms and the Descriptive Statistics for the sample means, the sample medians, the Z statistics, the t statistics and the Chi-square statistics.
Briefly discuss whether the shapes of the Histograms and the values of the Descriptive Statistics are or are not different from the results that you obtained in Question 1 (b). Are these new results consistent with what you were told in lectures about the distributions these statistics are supposed to have.
You should also use the appropriate tables to find the critical values for the z, t(n-1) and statistics associated with a probability of 0.05 in both the left hand tail and in the right hand tail. You should then check what percentage of your simulated values lie in these critical regions.
Is there any reason why your results might differ from what the theory says the results should be? (20 marks)
Total marks 25 = 5 + 20
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Question 3
When the desirable properties of sample estimators were discussed in lectures it was noted that either the sample mean or the sample median Md can be an unbiased estimator of the population mean m. In your answers you should comment on what difference (if any) it makes if we use the larger sample size of 50 instead of the smaller sample size of 10.
(a) Briefly state when we would expect that the sample mean or the sample median Md can both be unbiased estimators of the population mean m. (3 marks)
(b) Using the results that you have obtained in Questions 1 and 2 briefly discuss whether the sample mean and/or the sample median Md are unbiased estimators of the population mean m.. (4 marks)
(c) Using the results that you have obtained in Questions 1 and 2 briefly discuss whether the sample mean and/or the sample median Md are consistent estimators of the population mean m. (4 marks)
(d) Using the results that you have obtained in Questions 1 and 2 briefly discuss whether the sample mean is or is not a more relatively efficient estimator of the population mean m than the sample median Md is. (4 marks)
Total marks 15 = 3 + 4 + 4 + 4
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Question 4
The finance director of a large hardware firm is worried about the time that customers take to pay their accounts. He is also concerned about the differences in these times between city customers and country customers. He asks you to undertake a statistical study to help him to understand what is happening with the time that customers take to pay their accounts.
As a first step you take two random samples of the number of days that customers take to pay their accounts. The first random sample of 115 is taken from the list of city customers and we call these values X1. The second random sample of 85 is taken from the list of country customers and we call these values X2.
The finance director asks you to examine the times for the city customers or X1 values first. He then wants to compare the values for city customers with the times for country customers which we call the X2 values.
The X1 values can be found in cells A2:A116 and the X2 values can be found in cells C2:C86 in the Excel workfile Assignment 2 Data.xlsx.
He asks you to use the higher level of significance of a = 0.10 in all tests in this question.
In this question use Excel to find both the critical value and the p-value for each test.
Question 4 (contd.)
(a) Briefly explain why we use higher levels of significance like 0.10 or 0.15. (2 marks)
(b) Use Excel to obtain the descriptive statistics and an appropriate histogram for the values for the city customers. Briefly state what the histogram and the descriptive statistics obtained with Excel are telling us about the central values and the shape of the distribution. (5 marks)
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(c) In the past it has been found that the times that city customers take to pay their accounts have a population mean value of m = 34 and a population standard deviation of s = 6. Using the results that you have obtained in part (b) of this question test whether this mean has increased. Before you perform this test you should check whether the required assumptions are satisfied then when you perform the test use both the 6 step procedure and the appropriate p-value to choose between the appropriate hypotheses concerning the population mean m. (5 marks)
(d) The finance director thinks that the value of the standard deviation that was assumed to be known to be equal to 6 may have changed. He asks you to test whether the population standard deviation is different from 6. Before you perform this test for the value of the variance you should check whether the required assumptions are satisfied then when you test these hypotheses use the three possible approaches namely
(i) The 6 step procedure
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(ii) The appropriate p-value
(iii) The appropriate confidence interval. (5 marks)
(e) After completing the test in part (d) of this question the finance director now tests the hypotheses used in part (c) of the question using an appropriate testing statistic. When you perform this test you should check whether the required assumptions are satisfied and then use both the 6 step procedure and the appropriate p-value to choose between the hypotheses concerning the population mean m. (5 marks)
(f) In the past it has been found that the average time that country customers take to pay their accounts is up to 10 days more than the average time that city customers take to pay their accounts. After examining the most recent data however the Finance director comes to think that the average time taken by country customers is now more than 10 days longer than the average time the city customers take.
If it is assumed that the variance of the values for city customers is not different from the variance of the values for the country customers test the null hypothesis that the population means differ by 10 against the alternative hypothesis that the population means differ by more than 10.
Before you perform this test check the assumptions and then when you perform the test use both the 6 step procedure and the appropriate p-value to choose between the appropriate hypotheses concerning the difference between these population means.
.
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(5 marks)
Question 4 (contd.)
(g) The finance director now decides that you should test whether or not the variance of the values for city customers ( ) is the same as or different from the variance of the values for country customers ( ). Before you perform this test you should check whether the required assumptions are satisfied then when you test these hypotheses use the three possible approaches namely
(i) The 6 step procedure
(ii) The appropriate p-value
(iii) The appropriate confidence interval
After you complete this test for the values of the variances carry out the same test that you performed in part (f) and test the null hypothesis that the population means differ by 10 against the alternative hypothesis that the population means differ by more than 10. When you perform the test use both the 6 step procedure and the appropriate p-value to choose between the hypotheses concerning the difference between these population means. (8 marks)
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