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Chapter 7 Sampling Distributions & Confidence Intervals

Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer. Chapter 7 Sampling Distributions & Confidence Intervals. Doing Statistics for Business. Chapter 7 Objectives Motivation for Point Estimators Common Point Estimators

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Chapter 7 Sampling Distributions & Confidence Intervals

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  1. DoingStatistics for BusinessData, Inference, and Decision MakingMarilyn K. PelosiTheresa M. Sandifer Chapter 7 Sampling Distributions & Confidence Intervals

  2. DoingStatistics for Business Chapter 7 Objectives • Motivation for Point Estimators • Common Point Estimators • Desirable Properties of Point Estimators • Distribution of the Sample Mean: Large Sample or Known 

  3. DoingStatistics for Business Chapter 7 Objectives (con’t) • The Central Limit Theorem - A More Detailed Look • Drawing Inferences by Using the Central Limit Theorem • Large Sample Confidence Intervals for the Mean

  4. DoingStatistics for Business Chapter 7 Objectives (con’t) • Distribution of the Sample Mean: Small Sample and Unknown  • Small Sample Confidence Intervals for the Mean • Confidence Intervals for Qualitative Data • Sample Size Calculations

  5. DoingStatistics for Business Figure 7.1 Relationship between probability and inferential statistics

  6. Doing Statistics for Business A Point Estimate is a single number calculated from sample data. It is used to estimate a parameter of the population. A Point Estimator is the formula or rule that is used to calculate the point estimate for a particular set of data.

  7. DoingStatistics for Business TRY IT NOW! Sales of CD’s Comparing Point Estimators Calculated From Samples Selected From Different Populations Use a software package such as Excel or Minitab to simulate picking a sample of size n=10 from two different populations or use the samples given on the following slide.

  8. DoingStatistics for Business TRY IT NOW! Sales of Gizmos Comparing Point Estimators Calculated From Samples Selected From Different Populations (con’t) Store 1 Store 2 . 95 99 97 97 95 99 103 103 101 102 101 98 102 102 98 106 92 99 100 96

  9. DoingStatistics for Business TRY IT NOW! Sales of Gizmos Comparing Point Estimators Calculated From Samples Selected From Different Populations (con’t) Suppose the first variable is daily sales of a new CD at Store 1. The second variable is daily sales of the new CD at Store 2. Select a sample of size n=10 days from both stores. Assume that the days sales at both stores are normally distributed with a mean of 100 and a standard deviation of 3. Find and

  10. DoingStatistics for Business TRY IT NOW! Sales of CDs Comparing Point Estimators Calculated From Samples Selected From Different Populations (con’t) What do you notice about the difference in sample means even though the population means are the same? Find (Store 1 standard deviation squared), and What do you notice about the ratio of the two sample variances?

  11. Doing Statistics for Business An Unbiased Estimator yields an estimate that is fair. It neither systematically over- estimates the parameter nor systematically underestimates the parameter.

  12. DoingStatistics for Business TRY IT NOW! The Diaper Company Comparing the Variability of Two Point Estimators Weights in grams for the next 5 hourly samples taken at the diaper company are shown below: Hour 6: 54.89 55.06 55.45 55.23 55.75 Hour 7: 54.32 55.72 54.91 54.40 55.78 Hour 8: 54.14 55.18 55.78 55.37 55.69 Hour 9: 54.11 54.05 53.60 55.97 55.86 Hour 10: 55.21 55.40 53.87 55.09 55.70

  13. DoingStatistics for Business TRY IT NOW! The Diaper Company Comparing the Variability of Two Point Estimators (con’t) For each sample, calculate the sample mean and the sample median. Find the average of the sample means and the average of the sample medians. Find the standard deviation of the sample means and the standard deviation of the sample medians. Which point estimator has less variability?

  14. DoingStatistics for Business The Standard Error is the standard deviation of the sampling distribution of a point estimator. It measures how much the point estimator or sample statistic varies from sample to sample. The probability distribution of a point estimator or a sample statistic is called a Sampling Distribution.

  15. Doing Statistics for Business Central Limit Theorem (CLT) In random sampling from a population, with mean  and standard deviation , when n is large enough, the distribution of is • approximately normal with • a mean equal to  and • a standard deviation equal to

  16. Doing Statistics for Business Figure 7.2The Diaper Company Histogram of Individual Diaper Weights

  17. Doing Statistics for Business Figure 7.3The Diaper Company Histogram of 52 Sample Means

  18. Doing Statistics for Business Figure 7.4 Graphs of two populations with the same mean but different standard deviations

  19. Doing Statistics for Business Figure 7.5 Dotplots of Sample Means

  20. DoingStatistics for Business TRY IT NOW! The Central Limit Theorem Exploring the Third Point Use a software package such as Excel or Minitab to simulate picking 10 samples each of size n = 35 from two different populations: Population 1: Monthly sales of a leading on-line bookstore- normally distributed with a mean of  = 25 ($1,000) and a standard deviation of  = 1($1,000) Population 2: Monthly sales of a leading on-line bookstore- normally distributed with a mean of  = 25 ($1,000) and a standard deviation of  = 3 ($1,000)

  21. DoingStatistics for Business TRY IT NOW! The Central Limit Theorem Exploring the Third Point (con’t) For each sample, calculate a sample mean, . Find the average and standard deviation of the 10 ‘s from population 1 samples. Find the average and standard deviation of the 10 ‘s from population 2 samples.

  22. DoingStatistics for Business TRY IT NOW! The Central Limit Theorem Impact of Sample Size on Standard Error Using the random number table in Appendix A and the 350 values shown in the previous Try It Now! exercise in your textbook, select 10 samples of size 5 from population 1. Review Section 2.4 if you need a refresher on how to use the random number table. For each sample, calculate a sample mean, Find the average and standard deviation of the 10 ‘s. Compare these values to the corresponding values that you found in the previous Try It Now! for population 1. In that case the sample size was n = 35.

  23. Doing Statistics for Business Figure 7.6A Sampling Distribution of

  24. Doing Statistics for Business Discovery Exercise 1.1 The Central Limit Theorem in Action Part I. Draw a picture of a normal distribution with mean of 80 and standard deviation of 5. This is the population we will sample from.

  25. Doing Statistics for Business Discovery Exercise 1.1 The Central Limit Theorem in Action (con’t) Part II. Generate and examine 100 random samples. For this exercise you will need to generate 100 samples each consisting of 30 value selected from a normal distribution with a mean of 80 and a standard deviation of 5. Part III. Create a distribution of for samples of size n = 30.

  26. Figure 7.6B Sampling Distribution for when  = 55.00 Figure 7.6C Sampling Distribution for when  = 54.50 DoingStatistics for Business

  27. DoingStatistics for Business TRY IT NOW! Cost of Books Comparing The Sample Mean to the Claimed Population Mean A university states the average student spends $225 per semester on books. Based on your own experience you feel that this is an underestimate of the true expenditure. You ask 30 of your friends how much they spent on textbooks this past semester and you obtain the following data:

  28. DoingStatistics for Business TRY IT NOW! Cost of Books Comparing The Sample Mean to the Claimed Population Mean (con’t) 214 241 248 258 269 233 244 249 260 274 234 245 250 262 276 236 247 253 262 277 239 248 254 263 277 241 248 254 265 281 Based on these data, do you have reason to tell the university that its statement is inaccurate?

  29. Doing Statistics for Business A Confidence Interval or an Interval Estimate is a range of values with an associated probability or Confidence Level, 1 – . The probability quantifies the chance that you have an interval that contains the true population parameter.

  30. Figure 7.7. Normal Distribution with 0.05 in the tails. DoingStatistics for Business

  31. DoingStatistics for Business TRY IT NOW! The Bottle-Filling Problem A sample of 36 bottles had a sample mean of = 32.10 oz. The population standard deviation, , was assumed to be 0.1 oz. Find a 95% confidence interval for . How wide is the interval? Now find a 98% confidence interval for . Which interval is wider?

  32. Doing Statistics for Business Figure 7.8 Comparison of Confidence Intervals and µ

  33. Doing Statistics for Business Discovery Exercise 7.2 Exploring Confidence Intervals for  From a population of college students across the United States, a sample was selected to find our how many hours per week a typical student spends playing sports. Part I. A random sample of 2500 students was selected. The sample mean, was found to be 12.5 hours. The population standard deviation, , is known to be 1.05 hours. Given this information, find:

  34. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  (a) a 90% confidence interval for . (b) a 92% confidence interval for . (c) a 94%confidence interval for . (d) a 96% confidence interval for .

  35. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  (e) a 98% confidence interval for  (f) Discuss what happens to the size of the interval as the level of the confidence increases.

  36. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  Part II. A random sample of 2500 students was selected. The sample mean, was found to be 10.5 hours. The population standard deviation, , is known to be 1.05 hours. Given this information, find: (a) a 90% confidence interval for 

  37. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  (b) a 92% confidence interval for . (c)a 94%confidence interval for . (d) a 96% confidence interval for  (e) a 98% confidence interval for 

  38. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  Compare the intervals found in Part I with those found in Part II. Discuss what happened to the confidence interval due to the change in the value of the sample mean x. Part III. A random sample of 2500 students was selected. The sample mean, was found to be 12.5 hours. Suppose you that the population standard deviation, , is actually 2.05 hours. Given this information, find:

  39. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  (a) a 90% confidence interval for  (b) a 92% confidence interval for . (c)a 94%confidence interval for . (d) a 96% confidence interval for  (e) a 98% confidence interval for 

  40. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  Compare the intervals found in Part I with those found in Part III. Discuss what happened to the confidence intervals due to the change in the value of the population standard deviation, . Part IV. A random sample of 2000 students was selected. The sample mean, was found to be 12.5 hours. The population standard deviation, , is known to be 1.05 hours. Given this information, find:

  41. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  (a) a 90% confidence interval for  (b) a 92% confidence interval for . (c)a 94%confidence interval for . (d) a 96% confidence interval for  (e) a 98% confidence interval for 

  42. Doing Statistics for Business Discovery Exercise 7.2 (con’t) Exploring Confidence Intervals for  Compare the intervals found in Part I with those found in Part IV. Discuss what happened to the confidence interval due to the change in the value of the sample size, n.

  43. Doing Statistics for Business Figure 7.9t-distribution with 5 Degrees of Freedom

  44. Figure 7.10 t-distribution with 25 Degrees of Freedom Figure 7.11 t-distribution with 1 and 50 Degrees of Freedom DoingStatistics for Business

  45. Doing Statistics for Business Figure 7.12 A portion of the t table

  46. DoingStatistics for Business TRY IT NOW! Retirement Years Confidence Interval for A survey shows that a growing number of Americans are willing to make sacrifices to become home owners despite increasing job and financial worries. The Federal National Mortgage Association surveyed 1857 Americans and found that 67% would put off retirement for 10 years to own a home. Find a 90% confidence interval for the proportion of all Americans who would put off retirement for 10 years to own a home.

  47. DoingStatistics for Business TRY IT NOW! Bottle Filling Finding the Sample Size How many bottles does the bottle manufacturer need to sample to be 98% confident that the error is at most 0.002 oz? Remember that the population standard deviation is 0.1 oz.

  48. DoingStatistics for Business TRY IT NOW! Retirement Years Sample Size Calculation for  How many Americans must be sampled to determine the percentage who would put off retirement for 10 years to own a home? The estimate should not differ from the actual population proportion by more than 3% with a confidence of 90%.

  49. Doing Statistics for Business Finding Confidence Intervals Using KaddStat Instructions for small sample confidence interval for mean- all others are done similarly From the Kadd menu select Confidence Intervals>One Sample>Population Mean using t The dialog box opens.

  50. Doing Statistics for Business Finding Confidence Intervals Using Kadd(con’t)

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