10 1 Comparing Two Proportions Objectives SWBAT DESCRIBE

10. 1 Comparing Two Proportions Objectives SWBAT: • DESCRIBE the shape, center, and spread of the sampling distribution of the difference of two sample proportions. • DETERMINE whether the conditions are met for doing inference about p 1 − p 2. • CONSTRUCT and INTERPRET a confidence interval to compare two proportions. • PERFORM a significance test to compare two proportions.

• In 2010, Josh Hamilton had 45 hits in 166 at-bats vs left-handed pitching. This gave him an average of 0. 271 vs lefties. • Vs right-handed pitching Hamilton had 141 hits in 352 at-bats. This gave him an average of 0. 401 vs righties. • Does this data give us convincing evidence that Hamilton has a greater ability to get a hit vs right-handed pitchers as opposed to left-handed pitchers? • Up until now, we have only been examining one population to work with proportions and means. In chapter 10, we’ll begin examining two populations, and answer questions like the one formulated above.

What is meant by “the sampling distribution of the difference between two proportions? ”

The Sampling Distribution of the Difference Between Sample Proportions Choose an SRS of size n 1 from Population 1 with proportion of successes p 1 and an independent SRS of size n 2 from Population 2 with proportion of successes p 2. Note: The standard deviation is measuring how far the estimated difference in proportions will be from the true difference in proportions, on average.

b) Find the mean of the sampling distribution. Show your work. c) Find the standard deviation of the sampling distribution. Show your work. Because there at least 10(100) = 1000 cars in Nathan’s state and at least 10(70) = 700 cards in Kyle’s state, the standard deviation is:

What are the conditions for calculating a two-sample z interval for p 1 -p 2? Conditions For Constructing A Confidence Interval About A Difference In Proportions • Random: The data come from two independent random samples or from two groups in a randomized experiment. o 10%: When sampling without replacement, check that n 1 ≤ (1/10)N 1 and n 2 ≤ (1/10)N 2. •

This measures how far the difference in sample proportions will typically be from the difference in population proportions if we repeat the random sampling or random assignment many times.

What is the formula for a two-sample z interval for p 1 -p 2? Is this on the formula sheet? Remember the formula sheet only gives us the generic: Two-Sample z Interval for a Difference Between Two Proportions

Alternate Example: Gun Control Have opinions changed about gun control? Gallup regularly asks random samples of U. S. adults their opinion on a variety of issues. In a poll of 1011 U. S. adults in January 2013, 38% responded that they “were dissatisfied with the nation’s gun laws and policies, and want them to be stricter. ” In a similar poll of 1011 adults in January 2012, only 25% agreed with this statement. a) Explain why we should use a confidence interval to estimate the change in opinion rather than just saying that the percentage increased by 13 percentage points. Because of sampling variability, the difference of 0. 13 is unlikely to be correct b) Use the results of these polls to construct and interpret a 90% confidence interval for the change in the proportion of U. S. adults who would agree with the statement about gun laws.

Alternate Example: Gun Control Have opinions changed about gun control? Gallup regularly asks random samples of U. S. adults their opinion on a variety of issues. In a poll of 1011 U. S. adults in January 2013, 38% responded that they “were dissatisfied with the nation’s gun laws and policies, and want them to be stricter. ” In a similar poll of 1011 adults in January 2012, only 25% agreed with this statement. Plan: We should use a two-sample z interval for p 1 -p 2 if the conditions are met: Random: The data comes from random samples of U. S. adults 10%: There were more than 10(1011) = 10110 U. S. adults in 2013 and more than 10(1011) = 10110 U. S. adults in 2012 Large counts: Because all four values are at least 10, this condition is met. Note: The observed counts have to be whole numbers.

Alternate Example: Gun Control Have opinions changed about gun control? Gallup regularly asks random samples of U. S. adults their opinion on a variety of issues. In a poll of 1011 U. S. adults in January 2013, 38% responded that they “were dissatisfied with the nation’s gun laws and policies, and want them to be stricter. ” In a similar poll of 1011 adults in January 2012, only 25% agreed with this statement. Do:

Alternate Example: Gun Control Have opinions changed about gun control? Gallup regularly asks random samples of U. S. adults their opinion on a variety of issues. In a poll of 1011 U. S. adults in January 2013, 38% responded that they “were dissatisfied with the nation’s gun laws and policies, and want them to be stricter. ” In a similar poll of 1011 adults in January 2012, only 25% agreed with this statement. Conclude: We are 90% confident that the interval from 0. 0963 to 0. 1637 captures the true difference in the proportion of U. S. adults that were dissatisfied with the nation’s gun laws between 2013 and 2012. c) Based on the interval, is there convincing evidence that opinions about gun control have changed? Yes. 0 is not included in the interval, indicating it is not plausible that U. S. adults still feel the same way about gun control.

Can you use your calculator for the Do step? Are there any drawbacks? We already know this! No partial credit for the do step.

What are the conditions for conducting a two-sample z test for a difference in proportions? Conditions For Performing a Significance Test About A Difference In Proportions • Random: The data come from two independent random samples or from two groups in a randomized experiment. o 10%: When sampling without replacement, check that n 1 ≤ (1/10)N 1 and n 2 ≤ (1/10)N 2. • These are the same as for confidence intervals.

What standard error do we use for a 2 sample z test for a difference in proportions? What is the pooled (combined) sample proportion? Why do we pool the sample proportions? If H 0: p 1 = p 2 is true, the two parameters are the same. We call their common value p. We need a way to estimate p, so it makes sense to combine the data from the two samples. This pooled (or combined) sample proportion is: The standard error then becomes:

What is the test statistic for a two-sample z test for a difference in proportions? Is this on the formula sheet? What does the test statistic measure? • You get the generic formula (yay!) • It measures how far the difference in the sample proportions is from 0, in standardized units.

Alternative Example: Hearing loss Are teenagers going deaf? In a study of 3000 randomly selected teenagers in 198 -1994, 15% showed some hearing loss. In a similar study of 1800 teenagers in 2005 -2006, 19. 5% showed some hearing loss. (These data are reported in Arizona Daily Star, August 18, 2010). a) Do these data give convincing evidence that the proportion of all teens with hearing loss has increased?

Alternative Example: Hearing loss Are teenagers going deaf? In a study of 3000 randomly selected teenagers in 198 -1994, 15% showed some hearing loss. In a similar study of 1800 teenagers in 2005 -2006, 19. 5% showed some hearing loss. (These data are reported in Arizona Daily Star, August 18, 2010). a) Do these data give convincing evidence that the proportion of all teens with hearing loss has increased? Do: P-value:

Alternative Example: Hearing loss Are teenagers going deaf? In a study of 3000 randomly selected teenagers in 198 -1994, 15% showed some hearing loss. In a similar study of 1800 teenagers in 2005 -2006, 19. 5% showed some hearing loss. (These data are reported in Arizona Daily Star, August 18, 2010). a) Do these data give convincing evidence that the proportion of all teens with hearing loss has increased? b) Between the two studies, Apple introduced the i. Pod. If the results of the test are statistically significant, can we blame i. Pods for the increased hearing loss in teenagers? No. Since we didn’t do an experiment where we randomly assigned some teens to listen to i. Pods and other teens to avoid listening to i. Pods, we cannot conclude that i. Pods are the cause. It is possible that teens who listen to i. Pods also like to listen to music in their cars, and perhaps the car stereos are causing the hearing loss.

Is it OK to use your calculator for the Do step? Are there any drawbacks? Are we cereally still asking this question? ? ? Note: Make sure you use 2 -Prop. ZTest, not 2 -samp z test Note: X’s must be integers What mistake do students often make when defining parameters in experiments? How can you avoid it? • The use language that refers to the sample, such as the proportion who took… or the proportion of people in the group… • It is better to use present or future tense!

Alternate Example: Cash for quitters In an effort to reduce health care costs, General Motors sponsored a study to help employees stop smoking. In the study, half of the subjects were randomly assigned to receive up to $750 for quitting smoking for a year while the other half were simply encouraged to use traditional methods to stop smoking. None of the 878 volunteers knew that there was a financial incentive when they signed up. At the end of one year, 15% of those in the financial rewards group had quit smoking while only 5% in the traditional group had quit smoking. Do the results of this study give convincing evidence that a financial incentive helps people quit smoking compared to traditional methods? (These data are reported in Arizona Daily Star, February 11, 2009)

Alternate Example: Cash for quitters In an effort to reduce health care costs, General Motors sponsored a study to help employees stop smoking. In the study, half of the subjects were randomly assigned to receive up to $750 for quitting smoking for a year while the other half were simply encouraged to use traditional methods to stop smoking. None of the 878 volunteers knew that there was a financial incentive when they signed up. At the end of one year, 15% of those in the financial rewards group had quit smoking while only 5% in the traditional group had quit smoking. Do the results of this study give convincing evidence that a financial incentive helps people quit smoking compared to traditional methods? (These data are reported in Arizona Daily Star, February 11, 2009)

Alternate Example: Cash for quitters In an effort to reduce health care costs, General Motors sponsored a study to help employees stop smoking. In the study, half of the subjects were randomly assigned to receive up to $750 for quitting smoking for a year while the other half were simply encouraged to use traditional methods to stop smoking. None of the 878 volunteers knew that there was a financial incentive when they signed up. At the end of one year, 15% of those in the financial rewards group had quit smoking while only 5% in the traditional group had quit smoking. Do the results of this study give convincing evidence that a financial incentive helps people quit smoking compared to traditional methods? (These data are reported in Arizona Daily Star, February 11, 2009)

Alternate Example: Cash for quitters In an effort to reduce health care costs, General Motors sponsored a study to help employees stop smoking. In the study, half of the subjects were randomly assigned to receive up to $750 for quitting smoking for a year while the other half were simply encouraged to use traditional methods to stop smoking. None of the 878 volunteers knew that there was a financial incentive when they signed up. At the end of one year, 15% of those in the financial rewards group had quit smoking while only 5% in the traditional group had quit smoking. Do the results of this study give convincing evidence that a financial incentive helps people quit smoking compared to traditional methods? (These data are reported in Arizona Daily Star, February 11, 2009) Conclude: Since the p-value of approximately 0 is less than the significance level of 0. 05, we reject the null. We have convincing evidence that financial incentives help employees like these quit smoking.