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Introduction to Distributions Employability Skills As you complete this exercise, think about which of these employability skills you are using: Today’s Practical is in two parts. What will we cover in this Part? In this practical exercise we will: Examine the distribution of means from Normal populations. Examine the distribution of means from non-Normal populations. Explore the Central Limit Theorem for means. Saving your work Don’t forget that it is useful to save your work. Save your work to your storage device to retain a copy. IQs are normally distributed with a population mean of 100 and a population standard deviation of 15. The file IQ.xlsx contains five samples from this population. Download from iLearn and open IQ.xlsx Open the IQ.xlsx file and look at the data. The name of the worksheet is IQ Data . You will see five columns, each of length 100, titled Sample 1, Sample 2,… , Sample 5. Each column represents a random sample from a population with a mean of 100 (ie.  = 100) and a standard deviation of 15 (ie.  = 15). 1 | Introduction to Distributions Copyright Macquarie University 2020 Open the IQ data

Individuals data – summarising numerically and graphically Each of the five samples of IQ scores is stored in a separate column (A to E). We begin by obtaining descriptive statistics: Click Data and Data Analysis . Select Descriptive Statistics . Select all five columns of data. Select that you have Labels in First Row . Check Summary Statistics and New Worksheet . Then click OK . The numerical summaries for all selected columns should appear on a new worksheet. Give your new worksheet a meaningful name. Write down the mean and standard deviation for each sample (correct to 2dp): Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Mean ( 𝑦̅ ) 102.09 95.74 101.83 101.48 97.79 Standard deviation (s) 14.51 13.59 15.23 13.37 14.54 Are the sample means equal? No Are the sample means close to the population mean? Yes Are the sample standard deviations equal? No Are the sample standard deviations close to the population standard deviation? Yes Why are the samples different and why do they differ from their expected values? Every time we take a random sample from a population, the samples are likely to differ. The difference between samples is known as sampling variability or sampling error. If a sample is representative of the population and is large enough, the sample statistics (such as the mean and standard deviation) should be close to the population values. We will now produce a histogram of the data in Sample 1: Return to the IQ Data worksheet. Click Data and Data Analysis . Select Histogram . Select Input Range A1:A101 . Use the default bins . Select Labels . Check Chart Output and New Worksheet . Then click OK . Format the bars of the histogram to remove the gaps . Give your histogram a meaningful title and x-axis title . Give your worksheet a meaningful name. Sketch the shape of the histogram below: 2 | Introduction to Distributions Copyright Macquarie University 2020

How would you describe the shape of the histogram? The histogram appears unimodal and symmetric – an approximately Normal distribution. Why is this the result that you would expect to see? Samples should resemble the population from which they are taken. Since this sample came from a Normal population, the sample should follow a Normal distribution. Means data – creating and summarising So far the analysis we did above has created summaries of samples of individual values. Now we are going to create a column of means by calculating the mean of each row of data. We will calculate the mean for each row, using the random values from columns A to E to produce a column of means. These means come from samples of size n=5 because we have 5 values in each row. We will store the means that we calculate in column G. Return to the IQ Data worksheet. In cell G2, type =AVERAGE(A2:E2) Now we want to use the same function for cells G3 to G101. The easiest way to do this is to left click in cell G2. At the bottom right hand corner of the cell you can see a little square. Hover over that square until the Excel curser becomes a thin +. Using the left hand mouse button click and hold on the + and drag the curser down to G101. The formula should copy to cells Add a title in cell G1 of Row Means. Obtain descriptive statistics for the column of means: Click Data and Data Analysis . Select Descriptive Statistics . Select Column G . Select Labels in First Row . Check Summary Statistics and New Worksheet . Then click OK . Give your new worksheet a meaningful name. Produce a histogram for the column of means: Click Data and Data Analysis . Select Histogram . Select Input Range G1:G101 Use the default bins . Select Labels . Check Chart Output and New Worksheet . Then click OK . 3 | Introduction to Distributions Copyright Macquarie University 2020

Give your histogram a meaningful title and x-axis title . Give your new worksheet a meaningful name. Sketch the shape of the histogram below: How would you describe the shape of the histogram? The histogram appears unimodal and symmetric – an approximately Normal distribution. Now compare the summary statistics for Sample 1 in column A, with the row means you have calculated in column G. Find the sample statistics for column A and column G (correct to 1dp) and fill in the table below: IQ data Mean Media n SD Range Min Max Individuals (Column A: Sample 1) 102.1 101.7 14.5 72.1 64.6 136.7 Means (Column G: Row Means n=5) 99.8 99.6 5.5 26.9 86.2 113.1 Comment on the range of the data in Column G compared to Column A. Individuals had a range of 72.1 IQ points; from a minimum of 64.6 to a maximum of 136.7 IQ points. Mean IQ scores for samples of size 5 had a range of 26.9 IQ points; from a minimum of 86.2 to a maximum of 113.1 IQ points. How do the standard deviations of column A and column G compare? The standard deviation of Column A is 2.6 times the standard deviation of Column G. In lectures you learnt that when samples of size n are randomly selected from a population with mean  and standard deviation , then the distribution of the Sample Means has a mean of  and a standard error of . For the IQ data for individuals : the original population is normally distributed the population mean is  = 100 4 | Introduction to Distributions Copyright Macquarie University 2020

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