Section 7.2. Estimating a Population Proportion

Transcription

1 Section 7.2 Estimating a Population Proportion

2 Overview Section 7.2 Estimating a Population Proportion Section 7.3 Estimating a Population Mean Section 7.4 Estimating a Population Standard Deviation or Variance.

3 Overview Use sample data to estimate values of population parameters. Inferential Statistics Test hypotheses (or claims) made about population parameters.

4 Requirements The sample is a simple random sample. The conditions for the binomial distributions are satisfied. There are at least 5 success and at least 5 failures. np 5 nq 5

5 Key Concepts This section presents methods for using a sample proportion to make an inference about the value of the corresponding population proportion. Here are the three main concepts. Point Estimate Confidence Interval Sample Size

6 Point Estimate A single value (or points) used to approximate a population parameter. The sample proportion is the best point estimate of the population p. ^p ^p= x n

7 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Find the best point estimate of the population proportion p.

8 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Find the best point estimate of the population proportion p. ^p= x n = =0.530

9 Confidence Level The probability 1 α that the confidence interval actually does not contain the population parameter, assuming that the estimation process is repeated a large number of times. Also known as degree of confidence or the confidence coefficient.

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11 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Find α/2.

12 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Find α/2. α = = 0.05 α/2 = 0.025

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14 Margin of Error The maximum likely difference between the observed sample proportion and the true value of the population proportion. E=z α/2 ^p ^q n

15 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Identify the value of the margin of error, E.

16 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Identify the value of the margin of error, E. E=z α/2 ^p ^q n =z.025 (.53)(.47) 1002 =.031

17 Confidence Interval (CI) A range (or an interval) of values used to estimate the true value of a population parameter. Also known as interval estimate. ^p E< p< ^p+e ( ^p E, ^p+e)

18 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Construct the confidence interval

19 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Construct the confidence interval ( , ) (0.499, 0.561)

20 Interpreting a CI We are % confident that the interval from to actually does contain the true value of the population proportion p. This means that if we were to select many different samples of size n and construct the corresponding confidence intervals, the percentage of them would actually contain the value of the population proportion p.

21 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Write a statement that correctly interprets the confidence interval.

22 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Write a statement that correctly interprets the confidence interval. We have a 95% confidence that the interval from to actually does contain the true value of the population proportion

23 From a KRC Research poll in which 1002 respondents were asked if they felt vulnerable to identify theft, 531 said yes. Use a 95% confidence interval. Write a statement that correctly interprets the confidence interval. We have a 95% confidence that the interval from to actually does contain the true value of the population proportion

24 Sample n How to find the sample size required to estimate a population proportion. When the estimate is known n= [z α/2] 2 ^p ^q E 2 When no estimate is known n= [z α/2 ] E 2

25 An economist wants to know if the proportion of the US population who commutes to work via car-pooling is on the rise. What size sample should be obtained if the economist wants an estimate within 2 percentage points of the true population with 90% confidence if (a) the economist uses the 2006 estimate of 10.7% (b) obtained from the American Community Survey the economist does not use any prior estimates?

26 The economist uses the 2006 estimate of 10.7% obtained from the American Community Survey. E = 2% = 0.02 CI = 90% α = 0.10 Zα/2 = Z 0.05 = ^p = ^q = n= [1.645]2 (0.107)(.893) =

27 The economist does not use any prior estimates? E = 2% = 0.02 CI = 90% α = 0.10 Zα/2 = Z 0.05 = n=[1.645]2 (0.25) =1692