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Chapter 7: Confidence Intervals

Chapter 7: Confidence Intervals. The Review. Basic Idea. Instead of giving a point estimate for the parameter we are trying to estimate, we provide a range of values wherein the true value of the parameter is contained. This range of values is what we call confidence interval.

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Chapter 7: Confidence Intervals

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  1. Chapter 7: Confidence Intervals The Review

  2. Basic Idea • Instead of giving a point estimate for the parameter we are trying to estimate, we provide a range of values wherein the true value of the parameter is contained. • This range of values is what we call confidence interval. • When asked to calculate or construct a confidence interval, we need to end up with two values: the lower bound and the upper bound. • (1 – α) x 100% CI = (Lower bound, Upper bound)

  3. General Formula • For any parameter that we are interested in, the general formula for constructing a confidence interval is: where: CV = critical value (either a Z or T) SE = standard error of the point estimate

  4. Summary of Point Estimators

  5. Summary of Critical Values

  6. Summary of Standard Errors

  7. Z vs. t Interval • When constructing an interval for one population mean, when do you use Z? When do you use t? Answer: Look at the standard deviation: Population  use Z interval Sample  use t interval

  8. Examples Suppose a simple random sample of 15 students is drawn from a population of 3000 college students. Among sampled students, the average IQ score is 115 with a standard deviation of 10. What is the 90% confidence interval for the students' IQ score?

  9. I-clicker What is the parameter of interest? • One Population Proportion • Two-population mean Independent Samples • One Population Mean • Two-population mean dependent samples

  10. I-clicker For a 90% Confidence Interval, what is the critical value? • 1.345 • 1.761 • 2.145 • 1.645

  11. I-clicker For a 90% Confidence Interval, what is the standard error? • 115/15 • 115/sqrt(15) • 10/15 • 10/sqrt(15)

  12. Solution Point Estimate : Critical Value: Standard Error:

  13. Solution (Using TI-83/84) • Go to STAT -> Tests -> Tinterval The calculator should display this: Tinterval Inpt: Data Stats (since we have summary data, choose stats) 115 10 15 0.90 (confidence level expressed in decimal) Calculate

  14. Example 2 A major metropolitan newspaper selected a simple random sample of 1,600 readers and asked them whether the paper should increase its coverage of local news. Six hundred and forty wanted more local news. What is the 95% confidence interval for the proportion of readers who would like more coverage of local news? What is the parameter of interest?  One population proportion

  15. I-clicker What is the parameter of interest? • One Population Proportion • Two-population mean Independent Samples • One Population Mean • Two-population mean dependent samples

  16. I-clicker What is the point estimate? • 640 • 1600/640 • 640/1600 • 1600/sqrt(640)

  17. I-clicker What is the critical value for a 95% confidence interval? • 1.96 • 1.645 • 2.576 • 1.28

  18. I-clicker • What is the standard error? • 0.40/sqrt(1600) • 0.40/sqrt(640) • Sqrt(0.40*0.60)/1600 • Sqrt(0.40*0.60)/Sqrt(1600)

  19. Solution Point Estimate : Critical Value: Standard Error:

  20. Solution (Using TI-83/84) • Go to STAT -> Tests -> 1-PropZinterval The calculator should display this: 1-PropZInt x: n: C-level: Calculate

  21. Example 3 Suppose that simple random samples of college freshman are selected from two universities - 15 students from school A and 20 students from school B. On a standardized test, the sample from school A has an average score of 1000 with a standard deviation of 100. The sample from school B has an average score of 950 with a standard deviation of 90. What is the 90% confidence interval for the difference in test scores at the two schools, assuming that test scores came from normal distributions in both schools?

  22. I-clicker What is the parameter of interest? • One Population Proportion • Two-population mean Independent Samples • One Population Mean • Two-population mean dependent samples

  23. I-clicker What is the point estimate? • 100 – 90 • 1000 – 950 • 20 – 15 • (1000+950)/2

  24. I-clicker What is the degrees of freedom for a 95% confidence interval? • 19 • 14 • 34 • 33

  25. Solution Step 1: Construct a Table of given values

  26. Solution Step 2: Compute the Values needed for the interval Point Estimate : Critical Value: Standard Error:

  27. Solution

  28. Solution (Using TI-83/84) • Go to STAT -> Tests -> 2-SampTInterval The calculator should display this: 2-SampTInt Inpt: Data Stats (we want stats since we have summary data)

  29. Example 4 A corporate personnel manager is in charge of promoting the "wellness" of employees. One target is lowering blood pressure for employees who are under stress. The manager wants to test the effectiveness of a stress-reduction program designed to lower systolic blood pressure. Ten employees with high blood pressure were randomly selected. Their blood pressure was taken before and after participating in the stress-reduction program. Find a 95% confidence interval for the true difference in the systolic blood pressure of employees before and after the stress-reduction program.

  30. Example 4 Employee Before After Diff 1 158 148 10 2 176 133 43 3 150 152 -2 4 179 170 9 5 183 155 28 6 206 178 28 7 177 185 -8 8 165 151 14 9 175 180 -5 10 186 144 42

  31. I-clicker What is the parameter of interest? • One Population Proportion • Two-population mean Independent Samples • One Population Mean • Two-population mean dependent samples

  32. I-clicker What is the estimated average difference in blood pressure before and after the stress-reduction program? • 175.5 • 159.6 • 15.9 • 167.55

  33. I-clicker What is the standard error of the difference in averages? • 15.56/ sqrt(10) • 18.63/ sqrt(10) • 17.47/sqrt(10) • ((9*15.56)+(9*17.47))/28

  34. Solution Point Estimate : Critical Value: Standard Error:

  35. Solution (Using TI-83/84) • Input the data set into a list. • Compute for the difference between each observation and store it in a new list • Go to STAT -> Tests -> TInterval The calculator should display this: Tinterval Inpt: Data Stats (we want Data since we have data set) List: specify the list where the DIFFERENCES are stored Freq: keep as 1 C-Level: confidence level Calculate

  36. Sample Size Determination • Depending on the parameter that we want to estimate, there is an appropriate way to determine the sample size needed. Mean: Proportion:

  37. Example A fast food company wants to determine the average number of times that fast food users visit fast food restaurants per week. They have decided that their estimate needs to be accurate within plus or minus one-tenth of a visit, and they want to be 95% sure that their estimate does differ from true number of visits by more than one-tenth of a visit. Previous research has shown that the standard deviation is .7 visits. What is the required sample size?

  38. Solution

  39. Example A publishing wants to know what percent of the population might be interested in a new magazine on making the most of your retirement. Secondary data (that is several years old) indicates that 22% of the population is retired. They want to be 95% certain that their finding does not differ from the true rate by more than 5%. What is the required sample size?

  40. Solution

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