**Statistics Questions**

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**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

**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 s^{2} 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 s^{2} and check whether your answers agrees with what your obtained with the formulae.

(5 marks)

**Question 1 (contd.)**

(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 M_{d}

The Z statistic for the sample mean Z =

The unbiased estimate of the variance for each sample s^{2}.

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?

**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 M_{d}

The Z statistic for the sample mean Z =

The unbiased estimate of the variance for each sample s^{2}.

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?

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