Source: D. Morton, et al., Lead Absorption in Children of Employees in a Lead-Related Industry, American Journal of Epidemiology 155 (1982).

Transcription

1 STAT E-50 - Introduction to Statistics The Normal Model 1. Researchers have investigated lead absorption in children of parents who worked in a factory where lead is used to make batteries. Shown below are the levels of lead in the children s blood (in μg/dl of whole blood): Source: D. Morton, et al., Lead Absorption in Children of Employees in a Lead-Related Industry, American Journal of Epidemiology 155 (1982). a) Construct the histogram, using classes of 10-20, 20-30, etc: b) What would the data look like if we shift the data by subtracting 30 from each value? What happens to the center of the data? What happens to the spread?

2 c) What would the data look like if we rescale the data by dividing each of the original values by 15? What happens to the center of the data? What happens to the spread? 2. Here are some of the descriptive statistics for the original data and the shifted data: Descriptive Statistics N Range Minimum Maximum Mean Std. Deviation Statistic Statistic Statistic Statistic Statistic Std. Error Statistic leadlevel minus Valid N (listwise) 33 a) Which of these results changed when the data was shifted? By how much? b) Which results did not change?

3 3. Here are some of the descriptive statistics for the original data and the rescaled data: Descriptive Statistics N Range Minimum Maximum Mean Std. Deviation Statistic Statistic Statistic Statistic Statistic Std. Error Statistic leadlevel divby Valid N (listwise) 33 a) Which of these results changed when the data was rescaled? By how much? b) Which results did not change? 4. Here are the results when the data was shifted by subtracting the mean, and when the data was rescaled by dividing by the standard deviation: Descriptive Statistics N Range Minimum Maximum Mean Std. Deviation Statistic Statistic Statistic Statistic Statistic Std. Error Statistic leadlevel minusmean Valid N (listwise) 33 Descriptive Statistics N Range Minimum Maximum Mean Std. Deviation Statistic Statistic Statistic Statistic Statistic Std. Error Statistic leadlevel div.by.s Valid N (listwise) 33 a) What happened when the mean was subtracted? b) What happened when the data was divided by the standard deviation?

4 5. Here are the results when the data was standardized by subtracting the mean, and then dividing by the standard deviation: Descriptive Statistics N Range Minimum Maximum Mean Std. Deviation Statistic Statistic Statistic Statistic Statistic Std. Error Statistic leadlevel z Valid N (listwise) 33 The new mean is: The new standard deviation is: The z score: z y The z-score, or standard score, tells you how many standard deviations a value is from the mean.

5 The Nearly Normal Condition: The shape of the distribution is unimodal and symmetric. Check this condition by making a histogram or a Normal Probability Plot. The Normal Model: Normal models are appropriate for distributions whose shapes are unimodal and roughly symmetric. We represent a Normal model with mean of μ and standard deviation of σ as N(μ, σ). The Normal model with mean zero and standard deviation 1 is called the Standard Normal Model or the Standard Normal Distribution. We can write it N(0, 1). The Rule: In a Normal model, about 68% of the values fall within one standard deviation of the mean, about 95% of the values fall within two standard deviations of the mean, and about 99.7% of the values fall within three standard deviations of the mean.

6 6. Suppose that a test of coordination for first graders is scored so that the mean for all first graders is 50 and the standard deviation is 15. If we assume also that the distribution is unimodal and symmetric, a) what percent of first graders have a score that is between 20 and 50? b) what percent of first graders have a score that is between 35 and 65? c) If your first grade child scored an 80 on this test, how does your child stand among all children tested? d) A friend of yours claims that his daughter scored what he calls a perfect 100 on this test. Do you believe him? Why or why not?

7 e) What percent of students scored below 75? f) What percent of students scored below 30? g) What percent of students scored above 30? h) It is decided that students whose score is in the lowest 10% should receive special physical education training. What should be the cutoff score for this program?

8 The Normal Distribution: Using SPSS To find areas under the Normal curve, the data must first be converted to z-scores which will be added to the datafile. 1. Open the datafile you are using. For this illustration, we will use the Lead data. 2. Choose > Analyze > Descriptive Statistics > Descriptives 3. In the Descriptives dialog box, add the desired variable(s) to the list of variables to be analyzed, using the arrow. In our example shown below, the variable to be analyzed is named lead. Check the box labeled Save standardized values as variables to have SPSS compute the z scores for the selected variable(s) and save them in your datafile. Use the options button to specify the statistics you wish to include in the output. Click the OK button to generate the output and compute the z scores. You should see the new column Zlead in your datafile.

9 You can now find areas under the Normal curve below a specific value of z. 1. Choose > Transform > Compute Variable 2. Enter the name of the new variable in the Target Variable box. Select CDF & Noncentral CDF in the Function group box. Select Cdfnorm in the Functions and Special Variables box. Click on OK. (Note: CDF stands for Cumulative Distribution Function. Note the description provided in the box below the calculator functions.)

10 Go back to the Data View to see the results: Now you can use these results to answer some questions about the data. Try drawing a diagram What percent of the students had a lead level below 48? What percent of the students had a lead level above 21? What percent of the students had a lead level between 21 and 48?

11 To Create Normal Probability Plots Choose > Analyze > Descriptive Statistics > Q-Q Plots Move the variable you are analyzing to the Variables: box Be sure that the Test Distribution is specified as Normal Click on OK Here are the results for the lead level data: Page 11

12 ASSIGNMENT #3 - The Normal Model; Using SPSS Read Chapter 6 Exercises: p. 147: 2, 8, 12, 20, 22, 28, 34, 44, 46, 51 Use SPSS to do the following exercises: p. 78: 24, 42*, 46* p. 109: 32c* *For Exercises 42, 46, and 32c in addition to the display that is specified, also create any one of the following: dotplot, histogram, boxplot, stem and leaf display. Also: p. 150: Use the data from Exercise 31 to answer these questions: 1. Assuming the data is approximately normally distributed, a) What percent of the times are below seconds? b) What percent of the times are above seconds? c) What percent of the times are below seconds? d) What percent of the times are above seconds? 2. Construct a Normal Probability Plot and describe what you found. 3. Construct a Boxplot and discuss the shape of the distribution. 4. Do you think that the answers you found in Question 1 are valid? Why or why not? Be sure to include your SPSS results and graphs in your assignment. To copy a graph from SPSS to your Word document, - Right-click on the graph and select Copy Graph - Go to the appropriate location in your Word document and Paste it in. Important: In your Word document, right-click on the graph and make it smaller, but be sure that it is still clear and legible. Also, please be sure to Autofit any SPSS tables. It is recommended that you complete the ActivStats exercises related to Chapter 6, including the SPSS exercises, and the SPSS exercises for Chapters 1-5. Pay particular attention to the ActivStats activities noted in the textbook (Working with Normal Models, The Normal Table, Normal Models, etc.). Page 12