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Chapter 11

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Chapter 11

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  1. Chapter 11 Two-Sample Tests of Hypothesis

  2. Goals • Conduct a test of hypothesis about the difference between two independent population means when both samples have population standard deviation that are known (z) • Conduct a test of hypothesis about the difference between two independent population means when both samples have population standard deviation that are not known (t) • Conduct a test of hypothesis about the difference between two population proportions (z)

  3. Compare The Means From Two Populations • Is there a difference in the mean number of defects produced on the day and the night shifts at Furniture Manufacturing Inc.? • Comparing two means from two different populations • Is there a difference in the proportion of males from urban areas and males from rural areas who suffer from high blood pressure? • Comparing two proportions from two different populations

  4. Two Populations • Two Independent Populations • E-Trade index funds • Merrill Lynch index funds • Two random samples, two sample means • Mean rate of return for E-Trade index funds (10.4%) • Mean rate of return for Merrill Lynch index funds(11%) • Are the means different? • If they are different, is the difference due to chance (sampling error) or is it really a difference? • If they are the same, the difference between the two sample means should equal zero • “No difference”

  5. Two Populations • Two Independent Populations • Plumbers in central Florida • Electricians in central Florida • Two samples, two sample means • Mean hourly wage rate for plumbers ($30) • Mean hourly wage rate for electricians ($29) • Are the means different? • If they are different, is the difference due to chance (sampling error) or is it really a difference? • If they are the same, the difference between the two sample means should equal zero • “No difference”

  6. Distribution Of Differences In The Sample Means

  7. Theory Of Two Sample Tests: • Take several pairs of samples • Compute the mean of each • Determine the difference between the sample means • Study the distribution of the differences in the sample means • If the mean of the distribution of differences is zero: • This implies that there is no difference between the two populations • If the mean of the distribution of differences is not equal to zero: • We conclude that the two populations do not have the same population parameter (example: mean or proportion)

  8. Theory Of Two Sample Tests: • If the sample means from the two populations are equal: • The mean of the distribution of differences should be zero • If the sample means from the two populations are not equal: • The mean of the distribution of differences should be either: • Greater than zeroor • Less than zero

  9. Normal Distributions • Remember from chapter 9: • Distribution of sample means tend to approximate the normal distribution when n ≥ 30 • For independent populations, it can be shown mathematically that: • Distribution of the difference between two normal distributions is also normal • The standard deviation of the distribution of the difference is the sum of the two individual standard deviations

  10. Test Of Hypothesis About The Difference Between Two Independent Population Means (population standard deviation known) • Same five steps: • Step 1: State null and alternate hypotheses • Step 2: Select a level of significance • Step 3: Identify the test statistic (z) and draw • Step 4: Formulate a decision rule • Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses • Fail to reject null • Reject null and accept alternate • Assumptions & Formulas 

  11. FormulasTwo Independent Pop. Means (population standard deviation known) Standard Deviation of the Distribution of Differences • Assumptions: • Two populations must be independent (unrelated) • Population standard deviation known for both • Both distributions are Normally distributed

  12. Example 1: Comparing Two Populations • Two cities, Bradford and Kane are separated only by the Conewango River • The local paper recently reported that the mean household income in Bradford is $38,000 from a sample of 40 households. The population standard deviation (past data) is $6,000. • The same article reported the mean income in Kane is $35,000 from a sample of 35 households. The population standard deviation (past data) is $7,000. • At the .01 significance level can we conclude the mean income in Bradford is more?

  13. Example 1: Comparing Two Populations • We wish to know whether the distribution of the differences in sample means has a mean of 0 • The samples are from independent populations • Both population standard deviations are known

  14. Example 1: Comparing Two Populations • Step 1: State null and alternate hypotheses • H0: µB≤ µK , or µB= µK • H1: µB> µK • Step 2: Select a level of significance •  = .01 (one-tail test to the right) • Step 3: Identify the test statistic and draw • Both pop. SD known, we can use z as the test statistic • Critical value  .01 yields .49 area,  z = 2.33

  15. Example 1: Comparing Two Populations • Step 4: Formulate a decision rule • If z is greater than 2.33, we reject H0 and accept H1,otherwise we fail to reject H0 • Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses

  16. Example 1: Comparing Two Populations • Because 1.98 is not greater than 2.33, we fail to reject H0 • We can not conclude that the mean income in Bradford is more than the mean income in Kane • The p-value (table method) is: P(z ≥ 1.98) = .5000 - .4761 = .0239 • This is more area under the curve associated with z-score of 2.33 than for alpha • We conclude from the p-value that H0 should not be rejected • However, there is some evidence that H0 is not true

  17. Test Of Hypothesis About The Difference Between Two Independent Population Means (population standard deviation not known) • Same five steps: • Step 1: State null and alternate hypotheses • Step 2: Select a level of significance • Step 3: Identify the test statistic (t) and draw • Step 4: Formulate a decision rule • Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses • Fail to reject null • Reject null and accept alternate • Assumptions & Two Formulas 

  18. Assumptions necessary: • Sample populations must follow the normal distribution • Two samples must be from independent (unrelated) populations • The variances & standard deviations of the two populations are equal • Two Formulas • Pooled variance • T test statistic

  19. FormulasTwo Independent Population Means(population standard deviation not known) Degrees of Freedom = 2

  20. Pooled Variance • The two sample variances are pooled to form a single estimate of the unknown population variance • A weighted mean of the two sample variances • The weights are the degrees of freedom that form each sample • Why pool? • Because if we assume the population variances are equal, the best estimate will come from a weighted mean of the two variances from the two samples

  21. Example 2: Comparing Two Populations • A recent EPA study compared the highway fuel economy of domestic and imported passenger cars • A sample of 15 domestic cars revealed a mean of 33.7 MPG with a standard deviation of 2.4 MPG • A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of 3.9 • At the .05 significance level can the EPA conclude that the MPG is higher on the imported cars? • Assume: • Samples are independent • Population standard deviations are equal • Distributions for samples are normal

  22. Example 2: Comparing Two Populations • Step 1: State null and alternate hypotheses • H0: µD≥ µI • H1: µD< µI • Step 2: Select a level of significance •  = .05 (One-tail test to the left) • Step 3: Identify the test statistic and draw • Pop. SD not known, so we use the t distribution • df = 15 + 12 – 1 – 1 = 25 • One-tail test with  = .05 • Critical Value = -1.708

  23. Example 2: Comparing Two Populations • Step 4: Formulate a decision rule • If our t < -1.708, we reject H0 and accept H1, otherwise we fail to reject H0 • Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses • We must make the calculations for: • Pooled Variance • t-value test statistic

  24. Example 2:Step 5: Compute

  25. Example 2:Step 5: Conclude • Because -1.64 in not less than our critical value of -1.708, we fail to reject H0 • There is insufficient sample evidence to claim a higher MPG on the imported cars • The EPA cannot conclude that the MPG is higher on the imported cars

  26. Proportion • The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest • Example: • A recent survey of Highline students indicated that 98 out of 100 surveyed thought that textbooks were too expensive • The sample proportion is • 98/100 • .98 • 98% • The sample proportion is our best estimate of our population proportion

  27. Two Sample Tests of Proportions • We investigate whether two samples came from populations with an equal proportion of successes (U = M) • Assumptions: • The two populations must be independent of each other • Experiment must pass all the binomial tests

  28. Test Of Hypothesis About The Difference Between Two Population Proportions • Same five steps: • Step 1: State null and alternate hypotheses • Step 2: Select a level of significance • Step 3: Identify the test statistic (z) and draw • Step 4: Formulate a decision rule • Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses • Fail to reject null • Reject null and accept alternate • Two Formulas 

  29. Formulas (Two Sample Tests of Proportions)

  30. Example 3: Two Sample Tests of Proportions • Are unmarried workers more likely to be absent from work than married workers (m< u )? • A sample of 250 married workers showed 22 missed more than 5 days last year • A sample of 300 unmarried workers showed 35 missed more than 5 days last year •  = .05 • Assume all binominal tests are passed

  31. Example 3: Two Sample Tests of Proportions • Step 1: State null and alternate hypotheses • H0: m≥ u • H1: m< u • Step 2: Select a level of significance •  = .05 • Step 3: Identify the test statistic and draw • Because the binomial assumptions are met, we use the z standard normal distribution •  = .05  .45  z = -1.65 (one-tail test to the left) • Step 4: Formulate a decision rule • If the test statistic is less than -1.65, we reject H0 and accept H1, otherwise, we fail to reject H0

  32. Example 3: Two Sample Tests of Proportions • Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses

  33. Example 3: Two Sample Tests of Proportions • Step 5: Conclude • Because -1.1 is not less than -1.65, we fail to reject H0 • We cannot conclude that a higher proportion of unmarried workers miss more than 5 days in a year than do the married workers • The p-value (table method) is: • P(z > 1.10) = .5000 - .3643 = .1457 • .1457 > .05, thus: fail to reject H0

  34. Goals • Understand the difference between dependent and independent samples • Conduct a test of hypothesis about the mean difference between paired or dependent observations

  35. Understand The Difference Between Dependent And Independent Samples • Independent samples are samples that are not related in any way • Dependent samples are samples that are paired or related in some fashion

  36. Dependent Samples • The samples are paired or related in some fashion: • 1st Measurement of item, 2nd measurement of item • If you wished to buy a car you would look at the same car at two (or more) different dealerships and compare the prices (1 price, 2 price) • Before & After • If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program (1 weight, 2 weight)

  37. Distribution Of The Differences In The Paired Values • Follows Normal Distribution • If the paired values are dependent, be sure to use the dependent formula, because it is a more accurate statistical test than the formula 11-3 in our textbook • Formula 11-7 helps to: • Reduce variation in the sampling distribution • Two kinds of variance (variation between 1st & 2nd categories for paired values, &, variation between values) are reduced to only one (variation between 1st & 2nd categories for paired values) • Reduced variation leads to smaller standard error, which leads to larger test statistic and greater chance of rejecting H0 • DF will be smaller

  38. Test Of Hypothesis About The Mean Difference Between Paired Or Dependent Observations Make the following calculations when the samples are dependent: • where d is the difference • where is the mean of the differences • is the standard deviation of the differences • n is the number of pairs (differences)

  39. EXAMPLE 4 • An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis • At the .05 significance level can the testing agency conclude that there is a difference in the rental charged? • A random sample of eight cities revealed the following information

  40. Miami 41 39 Seattle 46 50 EXAMPLE 4 continued

  41. EXAMPLE 4 continued • Step 1: • Step 2: •  = .05 • Step 3: • Use t because n < 30, df = 7, critical value = 2.365 • Step 4: • If t < -2.365 or t > 2.365, H0 is rejected and H1 accepted, otherwise, we fail to reject H0

  42. Example 4 continued City Hertz Avis d d2 Atlanta 42 40 2 4 Chicago 56 52 4 16 Cleveland 45 43 2 4 Denver 48 48 0 0 Honolulu 37 32 5 25 Kansas City 45 48 -3 9 Miami 41 39 2 4 Seattle 46 50 -4 16 Totals 8 78

  43. Example 4 continued

  44. Example 4 continued • Step 5: • Because 0.894 is less than the critical value, do not reject the null hypothesis • There is no difference in the mean amount charged by Hertz and Avis

  45. Summarize Chapter 11 • Conduct a test of hypothesis about the difference between two independent population means when both samples have 30 or more observations • Conduct a test of hypothesis about the difference between two independent population means when at least one sample has less than 30 observations • Conduct a test of hypothesis about the difference between two population proportions • Understand the difference between dependent and independent samples • Conduct a test of hypothesis about the mean difference between paired or dependent observations