Math 243 Sections , 6.1 Confidence Intervals for ˆp

Size: px

Start display at page:

Download "Math 243 Sections , 6.1 Confidence Intervals for ˆp"

Edith Walker
5 years ago
Views:

1 Math 243 Sections , 6.1 Confidence Intervals for ˆp Overview Polls and Statistical Inference Confidence Levels and Critical z-values Standard Error and Margin of Error Confidence Intervals Statistical Polls Go to to find the poll on U.S. healthcare. Find the sample proportion who think ensuring healthcare is the government s responsibility. Click on the methodology link at the bottom of the article to find the sample size, and the margin of error. Then use the margin of error to write the confidence interval. Sample Proportion, pp : (point estimate) Margin of Error: Sample Size: Confidence Level: A confidence interval is the point estimate ± the margin of error for the given confidence level. It is more accurate to give a range for the population proportion rather than a point estimate. Confidence Interval: Statistical Inference Now that we have studied the sampling distribution of a proportion, ˆp, we can begin to look at inferential statistics. That is, we want to take a single sample and make an estimate of the population parameter, which we do not know. Cara Lee Page 1

2 Confidence Levels and Critical z-values How many standard deviations from the mean? Identify the critical z-scores for 99%, 95%, 90% and 80% confidence levels. 99% Confidence z* = 95% Confidence z* = 90% Confidence z* = 80% Confidence z* = Cara Lee Page 2

3 Standard Error The Standard Error (SE) is an estimate of the standard deviation of the sampling distribution of a proportion. It's used when we don't know the value of p and are not able to determine the true standard deviation. pp is the sample proportion or the proportion who answered yes in the sample. SSSS pp = pp qq nn Example 1. A survey of 2,000 hiring managers showed that 1,200 use social media sites to research job applicants. a. State the sample proportion and calculate the standard error for this sample proportion. b. Check the four conditions needed to use a sampling distribution for ˆp. Then draw and label the distribution. Based on the confidence levels we found above, we would expect 95% of random samples to fall within 1.96 standard deviations of the mean. Our margin of error is 1.96 times the standard error. Cara Lee Page 3

4 Margin of Error The Margin of Error (ME) is an estimate that expresses the amount of sampling error in the results of a survey. When you see something like ±3 percentage points, that is the margin of error. MMMM pp = ± zz SSSS pp or MMMM pp = ± zz pp qq Where z* is the critical z-score value that corresponds to the desired confidence level. nn c. Continuing the previous example, calculate the margin of error at the 95% confidence level. d. Write the confidence interval for the sample proportion. e. Interpret the confidence interval: We are % confident that the true proportion of is between % and %. Confidence Intervals If a point estimate follows the normal model with standard error SE, then a Confidence Interval for the population parameter is pp ± zz pp qq nn Where z* is the critical z-score value that corresponds to the desired confidence level. Critical z-scores: 99% Confidence zz = 95% Confidence zz = 90% Confidence zz = 80% Confidence zz = Cara Lee Page 4

5 What does 95% confidence mean? If we collected random samples over and over, with the same sample size: Each time we would get a different sample proportion, pp. From each pp, a different Standard Error, Margin of Error, and confidence interval would be computed. About 95% of these confidence intervals would capture the true proportion. About 5% would miss the true proportion. The true proportion, p Interpretation of Confidence Intervals We need to convey that the uncertainty is in the interval, not in the true proportion. That is why we use confidence rather than probability. Technically Correct: We are 95% confident that the interval from 58% to 62% captures the true proportion of hiring managers who use social media to research job applicants. More Casual, But Fine Incorrect We are 95% confident that between 58% and 62% hiring managers use social media to research job applicants. There is a 95% probability that the true proportion is between 58% and 62%. The true proportion is either in the interval or not. The randomness is not in the true proportion but in the confidence interval. Cara Lee Page 5

6 Example 2. A 2012 poll asked 166 adults whether they were baseball fans; 48% said that they were. a. Construct a 99% confidence interval for the true proportion of US adults that are baseball fans. b. Construct a 95% confidence interval for the true proportion of US adults that are baseball fans. c. Construct a 90% confidence interval for the true proportion of US adults that are baseball fans. d. Construct an 80% confidence interval for the true proportion of US adults that are baseball fans. Certainty vs. Precision As the confidence level increases, the interval gets. Why? Cara Lee Page 6

7 Determining Sample Size for a desired Margin of Error Using Algebra we can solve the margin of error formula for n: MMMM pp = ± zz pp qq nn The sample size needed to get a desired margin of error (ME) is given by the formula: nn = pp qq zz MMMM 2 Example 3. It's believed that 25% of adults over 50 never graduated high school. We wish to see if the same is true among 25 to 30 year olds. a. How many of this younger age group must we survey in order to estimate the proportion of non-grads to within 6% with 90% confidence? b. Suppose we want to cut the margin of error to 4% (again with 90% confidence). What's the necessary sample size? c. What is the relationship between the number of people sampled and the margin of error? Cara Lee Page 7

8 Practice 1. A 2016 Gallup poll asked 1021 U.S. adults whether they were satisfied with their current healthcare and 581 people said they were satisfied. a. Give a 95% confidence interval for the true proportion of U.S. adults who are satisfied with their healthcare. b. Explain what your interval means. Practice 2. An article titled Tongue Piercing May Speed Tooth Loss, Researchers Say" found that 18 out of 52 participants had receding gums, which can lead to tooth loss. a. How many people need to be surveyed in order to estimate the proportion of piercedtongue people with receding gums to within 3% with 95% confidence? b. Suppose we decide that a margin of error of 8% would be sufficient (again with 95% confidence). What's the necessary sample size? Cara Lee Page 8

Chapter 19. Confidence Intervals for Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Chapter 19. Confidence Intervals for Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc. Chapter 19 Confidence Intervals for Proportions Copyright 2010, 2007, 2004 Pearson Education, Inc. Standard Error Both of the sampling distributions we ve looked at are Normal. For proportions For means