Lesson Sampling Distribution of Differences of Two Proportions

Transcription

1 STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION The GPS software company, TeleNav, recently commissioned a study on proportions of people who text while they drive. The study suggests that there are differences in the texting-while-driving habits of men and women. 1 We will compare the proportions of male and female texters who text frequently while driving. Our primary tool for this comparison is subtraction. That sounds simple enough, but before we can really know when one of these differences between two proportions is significant, we must understand the nature of the distribution of such differences. This understanding allows us to apply the methods of inference we have learned margins of error, confidence intervals, and hypothesis tests to the comparisons between two population proportions. For any two distributions of sample proportions, the distribution of differences between sample proportions can be very large and difficult to picture. To ease the comprehension of what a distribution of differences looks like, we will turn to two very small populations of texting drivers. The first population consists of two males who text while driving and the second population consists of three females who text while driving. Within these populations, frequent texting while driving is defined to be at least seven texts while driving per week. Rarely texting is defined as less than seven texts per week while driving. These are represented as follows: F = frequently texting (at least seven texts while driving per week) R = rarely texting (less than seven texts while driving per week) From these small populations, we will collect sample proportions of drivers who text frequently. We will then create a distribution of differences between male and female proportions. This small distribution will help us find a way to quantify the center and spread of such distributions of differences. 1 TeleNav, Inc. (2010). TeleNav-commissioned survey suggests both genders have similar views on abiding by and breaking the rules of the road. Retrieved July 14, 2010, from

2 STATWAY STUDENT HANDOUT 2 TRY THESE As mentioned, the first population we will consider consists of two men who text while driving, where one of these men texts frequently. The observations are {Frequent, Rare}. This is a small population of one frequent texter out of two, with a population proportion of 1/2, which is represented as π = ½ = 0.5. Now construct a sampling distribution of sample proportions from this population. 1 Treating this collection of two texting men as a small population, we will compute all sample proportions from samples of size n = 2. Sampling is done with replacement to preserve the independence of the trials (so some samples will include the same observation twice). In the following table, all possible samples are listed. For the case the frequent texter is chosen twice, the proportion is 2/2 = 1.0. For the case where the first pick is a frequent texter but the second is not, the proportion is 1/2 = 0.5. A Fill in the remaining sample proportions of men who text frequently, each of which are denoted as p m. Sample Drawn With Replacement Sample Proportion, B We have previously given formulas for the mean, standard error, and variance of a sampling distribution of sample proportions. These formulas require the population proportion of men

3 STATWAY STUDENT HANDOUT 3 who frequently text while driving, (π = ½ = 0.50) and the sample sizes (n = 2). Evaluate these formulas for the current sampling distribution. 2 Next, suppose we have a population of three women who text while driving and only one of these women texts frequently. These are represented as {Frequent, Rare, Rare} A What is the population proportion of frequent texters in the population of women? B Compute all sample proportions from samples of women texters of size n = 2. Denote the sample proportion of frequently texting women as p w. Sample Drawn With Replacement Sample Proportion, 2/2 = 1.0 1/2 = 0.5 0/2 = 0.0 1/2 = /2 = 0.0

4 Proportions of Women Who Text Frequently STATWAY STUDENT HANDOUT 4 C For this collection of proportions from the women drivers whose population proportion is π w = 1/3 = 0.333, compute the mean, standard error, and variance. Remember that the samples are all of size n = 2. 3 Our goal is to discover the nature of the distribution of all differences (p m p w ). To accomplish this, consider the difference of every value of p m minus every value of p w. A In the top row of the following table, all men s proportions (p m ) found in Question 1b are listed. In the left column, all women s proportions (p w ) found in Question 2b are listed. Most of the differences (p m p w ) are listed as well, but seven are missing. In any cell where no difference is given, record the value of p m p w, where p m is recorded at the top of the cell s column and p w is recorded at the left of the cell s row. Proportions of Men Who Text Frequently The total number of differences is 9 4 = 36.

5 STATWAY STUDENT HANDOUT 5 B Fill in the frequency table with the frequencies and their corresponding probabilities as relative frequencies for the various values of p m p w. Record the probabilities as unreduced fractions whose denominators are each 36. The distribution of probabilities is the sampling distribution of differences. Frequency Probability Total 36 36/36 = 1 C Sketch a histogram of the probability distribution of p m p w on the graph below. The Sampling Distribution of Differences, 12/36 10/36 8/36 6/36 4/36 2/ This distribution gives probabilities for every possible difference between male proportions of frequent texters and female proportions of frequent texters. Is the distribution approximately symmetric? Is the distribution approximately bell-shaped?

6 STATWAY STUDENT HANDOUT 6 Recalling that the mean is a similar to a balancing point for a probability distribution, give a visual estimate of the mean. 4 Use technology to compute the mean and standard deviation of all differences summarized by the frequency table in Question 3 (or your instructor can provide these). A What is the mean of differences? Is this value close to the estimate made in Question 3c? B Compute the difference of the means of the men and women from Questions 1b and 2c. C Comparing your answers in Questions 4a and 4b, complete the statement below. The mean of differences in Question 4a is the of the means from Questions 1b and 2c. D Compute the variance of all differences (by squaring the standard deviation) summarized by the frequency table in Question 3b (or your instructor can provide this). E Compute the sum of the variances of men and women texters from Questions 1b and 2c. F Comparing your answers in Questions 4d and 4e, complete the following statement. The variance of differences in Question 4d is the of the variances from Questions 1b and 2c. G Compute the standard deviation (or error) of all differences of proportions by taking the square root of the variance from Question 4d. Round this answer to the nearest tenth.

7 STATWAY STUDENT HANDOUT 7 H Now that you have quantified the mean and standard deviation (or error) of differences, you can begin to make inferences regarding extreme values. Do you consider a difference of p m p w = 1.0 to be unusually low? Why? Do you consider a difference of p m p w = 0.5 to be unusually high? Why? NEXT STEPS For this part of the lesson, we will expand our simulated populations of men and women using decks of cards. Your instructor will provide a deck of black cards and a deck of blue cards. These decks represent the following. The black deck is the population of males who text while driving. The blue deck is the population of females who text while driving. Do not count the cards in the decks! Just as the exact truth is always unknown in large populations, the truth must remain unknown with the decks of cards. In this way, you can experience the uncertainty that occurs when working with samples. Within each deck, there are red cards and black cards. The color of each card s front side represents the following: A card with a red front is a person who sends texts frequently while driving. A card with a black front is a person who sends texts rarely while driving. Your instructor has manipulated the decks of cards so that the proportions of red cards in each deck roughly match the proportions of men and women drivers who text frequently in the real world.

8 STATWAY STUDENT HANDOUT 8 TRY THESE 5 From each deck of cards, you and a partner will sample 10 cards, randomly replacing each card after it is drawn before drawing the next card to preserve the independence of the trials. A Shuffle the black deck (representing the men who text while driving) and draw 10 cards with replacement. Count the red front cards drawn. What is the sample proportion of frequent texters for the men (the number of red front cards divided by the sample size n = 10)? The dotplot below includes a simulation of 30 additional proportions (p m ) generated by drawing cards from a deck, just as you have done. Add your data point to this dotplot. Men who Text Frequently Proportions of Men who Text Frequently While Driving (Seven or More Texts Per Week) Men Proportions Population includes only men who admit to texting while driving. B Shuffle the blue deck (representing the women who text while driving) and draw 10 cards with replacement. Count the red front cards drawn. What is the sample proportion of frequent texters for the women (the number of red front cards divided by the sample size n = 10)?

9 STATWAY STUDENT HANDOUT 9 Once again, a dotplot is provided below which includes 30 proportions (p w ) generated in the same way. Add your proportion to this dotplot. Women who Text Frequently Proportions of Women who Text Frequently While Driving (Seven or More Texts Per Week) Women Proportions Population includes only women who admit to texting while driving. C Compute the difference between the proportions you found in Questions 5a and 5b. The dotplot below is generated by taking differences between the sample proportions p m and p w in the previous dotplots. Add your difference to the dotplot of differences below. Differences Between Men and Women Proportions For Frequent Texters (Seven or More Texts Per Week) Differences Populations include only people who admit to texting while driving.

10 STATWAY STUDENT HANDOUT 10 D Add any additional differences (p m p w ) generated by others in your class to this dotplot. From the dotplot of texting men, estimate the mean by picking one proportion that is representative of the group s center. E From the dotplot of texting women, estimate the mean by picking one proportion that is representative of that group s center. F The difference of means is the mean of the differences (p m p w ). Thus, an estimate of the mean difference can be found by subtracting the estimated means from Questions 5d and 5e. Subtract these values. Does this value make a good representation of the center of the dotplot of differences? G Using complete sentences, refer to the dotplot in Question 5c to describe the distribution of differences (p m p w ) in terms of shape, symmetry, and center. 6 The variances of sample proportions (p m and p w ) are each computed as the square of the standard error, When the population proportions (π) are unknown, estimate them using sample proportions p m and p w ). The variance of the differences (p m p w ) is the sum of the individual variances. This variance is therefore estimated as a sum,

11 STATWAY STUDENT HANDOUT 11 Applying a square root gives an estimate of the standard error of differences (p m p w ). A Use your representative values for p m and p w (from Questions 5d and 5e) to estimate the standard error of differences between all such proportions. B Using the estimated mean of p m p w values (from Question 5f) and the standard error (from Question 6a), do you consider a difference of p m p w = 0.6 to be unusually high? C Using the estimated mean of the representative value of p m p w (from Question 5f) and the standard error (from Question 6a), do you consider a difference of p m p w = 0 to be unusually low? D Suppose that the mean value of p m is the population proportion, π m, of men who text successively. Suppose also that the mean value of p w is the population proportion, π w, of women who text successively. What is implied by the approximate difference between π m and π w given in your answer from 6c above?

12 STATWAY STUDENT HANDOUT 12 TAKE IT HOME To compare the proportion of men who are left-handed to the proportion of women who are left-handed, twenty-five samples, each containing 575 men each, were gathered. From each of these samples of men, a sample proportion, p m, of those who are left-handed was computed. Additionally, twenty samples of 815 women each were gathered, and from each, a sample proportion, p w, of women who are left-handed was computed. Thus, twenty-five sample proportions, p m, of left-handed men, and twenty sample proportions, p w, of lefthanded women were gathered. From these, the collection of all possible differences, p m p w, was constructed. The distribution of differences is plotted below. Differences of Left Handed Proportions For Men and Women Male Proportion Minus Female Proportion Twenty-five samples of men gathered with 575 members in each. Twenty Samples of women gathered with 815 members each. 1 Do you consider an 8% difference between the proportions of men and women who are lefthanded likely? If unlikely, is the difference too high or too low? 2 If you assume an 8% difference, is the proportion of left-handedness higher for men or women?

13 STATWAY STUDENT HANDOUT 13 3 Do you consider a 1% difference likely? If unlikely, is the difference too high or too low? 4 If you assume a 1% difference, is the proportion of left-handedness higher for men or women? 5 Pick a difference that you consider a representative value of the differences on the dotplot. 6 Does your chosen difference allow for p m and p w to be equal? If not, then which is greater? 7 Suppose that a random proportion of left-handedness for men, p m = 0.10, is chosen from a sample of size n m = 575. Suppose also that a random proportion of left-handedness for women, p w = 0.08, is chosen from a sample of size n w = 815. Estimate the standard error in the differences of sample proportions, p m p w, rounded to two places after the decimal. 8 Estimate, roughly, the number of standard errors that the difference, p m p w = 0.08, lies from your representative difference in Question 5. Does the estimated value support your answer from Question 1? 9 Estimate, roughly, the number of standard errors that the difference, p m p w = 0.01, lies from your representative difference in Question 5. Does the estimated value support your answer from Question 3?

14 STATWAY STUDENT HANDOUT This lesson is part of STATWAY, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel STATWAY and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.