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## Chapter 11

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**Chapter 11**Two-Sample Tests of Hypothesis**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)**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**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”**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”**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)**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**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**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 **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**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?**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**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**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**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**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 **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**FormulasTwo Independent Population Means(population standard**deviation not known) Degrees of Freedom = 2**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**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**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**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**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**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**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**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 **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**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**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**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**Goals**• Understand the difference between dependent and independent samples • Conduct a test of hypothesis about the mean difference between paired or dependent observations**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**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)**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**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)**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**Miami**41 39 Seattle 46 50 EXAMPLE 4 continued**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**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**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**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