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Confidence Intervals for Proportions

Confidence Intervals for Proportions. Chapter 19. Overview and Objectives. This chapter presents the beginning of inferential statistics . The two major applications of inferential statistics involve the use of sample data to: Estimate the value of a population parameter

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Confidence Intervals for Proportions

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  1. Confidence Intervals for Proportions Chapter 19

  2. Overview and Objectives • This chapter presents the beginning of inferential statistics. • The two major applications of inferential statistics involve the use of sample data to: • Estimate the value of a population parameter • Example – Your local newspaper polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. Create a confidence interval to estimate the actual sentiment of all voters (i.e. the proportion of the population that supports the school budget).

  3. Overview and Objectives (cont.) • Major applications of inferential statistics (con’t): • 2.) Test some claim (or hypothesis) about a population parameter. • Example – In a recent year, of the 109,857 arrests for Federal offenses, 29.1% were for drug offenses. Test the claim that the drug offense rate is equal to 30%.

  4. Overview and Objectives (cont.) • In Chapter 19, our goal is to estimate a population proportion. Our specific objectives will be to: • Define a proportion. • Identify the assumptions and conditions necessary for estimating a population proportion. • Construct a confidence interval which serves as our estimate of the population proportion. • Calculate the specific sample size needed for our desired level of precision and confidence.

  5. What Is a Proportion? Proportion – is the fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest. Example: A recent survey indicated that 137 out of 1000 students with credit cards reported debits in excess of $500. The sample proportion is 137/1000, or 0.137, or 13.7 percent. If we let represent the sample proportion, X the number of “successes,” and n the number of items sampled, we can determine a sample proportion as follows.

  6. A Few Definitions A point estimate is a single value (or point) used to approximate a population parameter. The sample proportion is the best point estimate of the population proportion p. A confidence interval is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.

  7. A Confidence Interval Example– 1998 General Social Survey Question – do you agree or disagree with the following statement: “It is more important for a wife to help her husband’s career than to have one herself.” Response: 19% of 1823 respondents agreed. Create a C.I. to estimate percentage of Americans who would agree with this statement. Recall that the sampling distribution model of is centered at p, with standard deviation . Since we don’t know p, we can’t find the true standard deviation of the sampling distribution model, so we need to find the standard error:

  8. A Confidence Interval (cont.) • By the 68-95-99.7% Rule, we know: • about 68% of all samples will have a within 1 SE of p • about 95% of all samples will have a within 2 SEs of p • about 99.7% of all samples will have a within 3 SEs of p • We can look at this from ’s point of view…

  9. A Confidence Interval (cont.) • Consider the 95% level • There is a 95% chance that p is no more than 2 SEs away from • So, if we reach out 2 SEs, we are 95% sure that p will be in that interval. In other words, if we reach out 2 SEs in either direction of , we can be 95% confident that this interval contains the true proportion. • This is called a 95% confidence interval

  10. What Does “95% Confidence” Really Mean? Each confidence interval uses a sample statistic to estimate a population parameter. But, since samples vary, the statistics we use, and thus the confidence intervals we construct, vary as well. The figure on the next slide shows that some of our confidence intervals capture the true proportion (the green horizontal line), while others do not.

  11. What Does “95% Confidence” Really Mean? (cont.) Confidence intervals from ten different samples. Assume the true population proportion, p, is 0.20. P=0.20 Thus, we expect 95% of all 95% confidence intervals to contain the true parameter that they are estimating. This interval does not contain 0.21

  12. Margin of Error • We can claim, with 95% confidence, that the interval contains the true population proportion. • The extent of the interval on either side of is called the margin of error (E). • In general, confidence intervals have the form estimate ± E. • The more confident we want to be, the larger our ME needs to be.

  13. Critical Value • The ‘2’ in (our 95% confidence interval) came from the 68-95-99.7% Rule. • Using the z-scores table, we find that a more exact value for our 95% confidence interval is 1.96 instead of 2. • We call 1.96 the critical value and denote it z*. • For any confidence level, we can find the corresponding critical value.

  14. Critical Values (cont.) Example: For a 90% confidence interval, the critical value is 1.645:

  15. Assumptions and Conditions • Here are the assumptions and the corresponding conditions you must check before creating a confidence interval for a proportion: • Independence Assumption: The data values are assumed to be independent from each other. We check three conditions to decide whether independence is reasonable. • Plausible Independence Condition: Is there any reason to believe that the data values somehow affect each other? This condition depends on your knowledge of the situation—you can’t check it with data.

  16. Assumptions and Conditions (cont.) • Independence Assumption (con’t): • Randomization Condition: Were the data sampled at random or generated from a properly randomized experiment? Proper randomization can help ensure independence. • 10% Condition: Is the sample size no more than 10% of the population? Sample Size Assumption: The sample needs to be large enough for us to be able to use the CLT. • Success/Failure Condition: We must expect at least 10 “successes” and at least 10 “failures.”

  17. Estimating a Population Proportion Confidence interval for the population proportion (p) The confidence interval is often expressed in the following equivalent formats: or

  18. Creating a Confidence Interval for a Proportion - Step - by - Step! • Based on the General Social Survey results that indicated 19% of 1823 respondents agreed with the following statement :“It is more important for a wife to help her husband’s career than to have one herself”, estimate the proportion of Americans who would support this statement. • Step 1: Think • Identify the parameter you wish to estimate. • Identify the population about which you wish to make statements. • Choose and state a confidence level. • We want to find an interval that is likely with 90% confidence to contain the true proportion, p, of adults who believe a woman should sacrifice her career to support her husband.

  19. Creating a Confidence Interval for a Proportion - Step - by - Step! • Step 2: Check the conditions • Plausible Independence Condition - It is very unlikely that any of their respondents influenced each other. • Radomization Condition – The General Social Survey used a random sample of adults. • 10% Conditions - Although sampling was necessarily without replacement, there are many more U.S. adults than were sampled. The sample is certainly less than 10% of the population.

  20. Creating a Confidence Interval for a Proportion - Step - by - Step! • Step 3: Construct the confidence interval • A.) Find the critical value for a 90% confidence • level. - 1.645 • B.) Find the standard error. • C.) Find the margin of error. • E = Z* *SE(^p) = 1.645*0.009 = 0.0148 • D.) Write the confidence interval. • 0.19+ 0.015 or (0.205, 0.175)

  21. Creating a Confidence Interval for a Proportion - Step - by - Step! • Step 4: Conclusion • Interpret the confidence interval in the proper context. We are 90% confident that our interval captured the true proportion. • We are 90% confident that between 17.5% and 20.5% of all U.S. adults believe women should sacrifice their own careers to support the careers of their husbands.

  22. Interpretation of the Confidence Interval Don’t Misstate What the Interval Means: • Don’t suggest that the parameter (p) varies. • Don’t claim that other samples will agree with yours. (If use a different to create the interval the solution will be different.) • Don’t be certain about the parameter. (Be sure to include a statement about the level of confidence in your interpretation)

  23. Effects of a Higher Confidence Level • If we wanted to be 95% confident, how would our confidence level be affected? • To be 95% confident, we need to change our critical value to 1.96. Which would result in a new margin of error. • E = Z* *SE( ) = 1.96*0.009 = 0.018 • Therefore our new confidence interval is 0.19+ 0.018 or (0.172, 0.208). Our interval has become wider.

  24. Margin of Error: Certainty vs. Precision Our margin of error was 1.8%. If we wanted to reduce it to + 1.5% would our level of confidence be higher or lower? In general, confidence intervals have the form estimate± E. The more confident we want to be, the larger our margin of error needs to be. To be more confident, we wind up being less precise. We need more values in our confidence interval to be more certain. Because of this, every confidence interval is a balance between certainty and precision.

  25. Effects of a Larger Sample Size If the General Social Survey had polled more people, say 2000, would the intervals margin of error have been larger or smaller? New margin of error: Increasing the sample size reduces the variability in the sampling distribution of the sample proportion which results in a smaller standard error.

  26. Choosing Your Sample Size In general, the sample size needed to produce a confidence interval with a given margin of error at a given confidence level is: where za/2 is the critical value for your confidence level. To be safe, round up the sample size you obtain.

  27. Choosing Your Sample Size What sample size does it take to estimate the outcome of this survey with 95% confidence and a margin of error of 3%? We will do what polling organizations usually do and choose the most cautious proportion, 50%. We need at least 1,068 respondents to keep the margin of error as small as 3% with a confidence level of 95%

  28. Example – Legalizing Marijuana • In August 2000, the Gallup Poll asked 507 randomly sampled adults the question “Do you think the possession of small amounts of marijuana should be treated as a criminal offense?” Of these, 51% said Yes, 47% responded No, and 2% said they didn’t know. • A.) Create a 95% confidence interval for the percentage of adults who support the legalization of marijuana (i.e. those who would respond “No” to this question. Interpret your findings. • B.) Would a referendum proposing the legalization of marijuana likely pass? Explain.

  29. Example – Network TV The Fox TV network is considering replacing one of its prime-time crime investigation shows with a new family-oriented comedy show. Before a final decision is made, network executives commission a random sample of 400 viewers. After viewing the comedy, 250 indicated they would watch the new show and suggested it replace the crime investigation show. A.) Estimate the value of the population proportion. B.) Compute the standard error of the proportion. C.) Develop a 99 percent confidence interval for the population proportion. D.) Interpret your findings.

  30. Example – Environmental Protection • In 2000, the GSS asked participants whether they would be willing to pay much higher prices in order to protect the environment. Of n = 1154 respondents, 518 indicated a willingness to do so. A.) Find a 95% confidence interval for the population proportion of adult Americans willing to do so at the time of that survey. B.) Interpret that interval.

  31. Assignment • Read Chapter 20: Testing Hypotheses About Proportions • Try the following exercises from Ch. 19: • #1, 3, 5, 7, 9, 13, 23, 31, 33, 37

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