Exam 11: Hypothesis Tests Involving Two Sample Means

Politician In an effort to test the hypothesis that the proportion of voters in the younger than 40 year old age bracket who will vote for a particular politician is different than the proportion voters in the above 40 age bracket,the following data was collected. Papulation Number Wha Will Vate far Palitician Sample Size Below 40 348 700 Above 40 290 650 -Use $\alpha$ = 0.10.State your conclusions in terms of the test statistic. What is the p-value?

Battery Manufacturer A battery manufacturer claims that their new "long-lasting" battery has an average life that is significantly longer than their competitor's "regular" battery.Both types of batteries were tested and the number of hours each battery lasted is shown in the following table. Long-Leasting Battery (Pupulation 1) Regular Battery (Pupulation 2) 51 42 44 29 58 51 36 38 48 39 53 44 57 35 40 40 49 48 44 45 60 50 -Assume unequal variances.Using $\alpha$ = 0.01,state your conclusions.

SAT A new SAT preparation program claims that students will improve their verbal scores with a practice test after one month on the program.To validate their claim,they recorded the starting and practice verbal SAT scores of 12 individuals after a one-month period.The results are shown in the following table. Starting SAT Score Practice SAT Score 450 460 390 390 425 430 510 500 520 525 440 420 550 570 610 625 490 510 475 485 535 540 500 500 -Use $\alpha$ = 0.01.State the decision rule in terms of the critical value.

Tests in which samples are not independent are referred to as ______________________________.

Assume two normally distributed populations with equal population standard deviations.A random sample is taken from each population.Determine the standard error of the estimate for the following situation: 16 14 8 5

Medicare An economist wishes to test whether there is any difference in the proportion of households receiving Medicare for white households with less than $5,000 income per year and white households with $15,000 or more income per year.A random sample from households with less than $5,000 income per year is taken,and a random sample of households with $15,000 or more income is taken.The results of the samples are: Less than \ or more Sample size 200 150 Medicare Recipients 157 139 -Use a 0.01 significance level to determine if there is sufficient evidence to conclude that a difference in proportions exists.What is the random variable?

A typical example of independent samples occurs with before-and-after measurements for the same individuals or test units.

Chrysler corp Chrysler Corp.is responding to an advertisement by Toyota,which claims that its compact version gets much better gasoline mileage for in-city driving than any U.S.made car.Below are the results of surveys conducted to test the claim.Perform the appropriate test at the 0.01 significance level. Toyota Clurysler =30 =30 =24 =22.9 =91.28 =169.3 -State the null and alternative hypotheses. H₀: ____________________ H₁: ____________________

Medicare An economist wishes to test whether there is any difference in the proportion of households receiving Medicare for white households with less than $5,000 income per year and white households with $15,000 or more income per year.A random sample from households with less than $5,000 income per year is taken,and a random sample of households with $15,000 or more income is taken.The results of the samples are: Less than \ or more Sample size 200 150 Medicare Recipients 157 139 -Use a 0.01 significance level to determine if there is sufficient evidence to conclude that a difference in proportions exists.What are the null and alternative hypotheses? H₀: ____________________ H₁: ____________________

A university vice president for academic affairs randomly selects 15 tenured faculty from the College of Education and 15 tenured faculty from the College of Business.Each faculty member is then asked to rate his or her job satisfaction on a scale of 1(very dissatisfied)to 10(very satisfied).Would this be an example of independent samples or dependent samples? Explain.

When the population standard deviations are unknown and are not assumed to be equal,s₁ and s₂ must be used to estimate their respective population standard deviations, $\text { ? }$ ₁ and $\text { ? }$ _2.

In testing the difference between two population means using two independent samples,the population standard deviations are assumed to be unknown,each sample size is ? 30,and the calculated test statistic z = 2.56.If the test is two-tail and 5% level of significance has been specified,the conclusion should be to:

SAT A new SAT preparation program claims that students will improve their verbal scores with a practice test after one month on the program.To validate their claim,they recorded the starting and practice verbal SAT scores of 12 individuals after a one-month period.The results are shown in the following table. Starting SAT Score Practice SAT Score 450 460 390 390 425 430 510 500 520 525 440 420 550 570 610 625 490 510 475 485 535 540 500 500 -Use $\alpha$ = 0.01.State your conclusions.

The z-test approximation to the unequal variance t-test assumes that the population standard deviations, $\text { ? }$ ₁ and $\text { ? }$ ₂ ,are known.

Chrysler corp Chrysler Corp.is responding to an advertisement by Toyota,which claims that its compact version gets much better gasoline mileage for in-city driving than any U.S.made car.Below are the results of surveys conducted to test the claim.Perform the appropriate test at the 0.01 significance level. Toyota Clurysler =30 =30 =24 =22.9 =91.28 =169.3 -What is the conclusion?

1000 companies From the top 1000 companies in valuation,a comparison of the mean dividend yield of the top and bottom ten electric utility companies is desired.Let 1 = top ten,and 2 = bottom ten,electric utility companies respectively.The data shown below are for two independent samples,which are drawn at random from populations that are normally distributed. Top ten Bottom ten 8.76 6.65 8.96 6.38 7.20 6.24 8.92 6.93 8.61 7.46 7.81 0.00 6.58 7.78 8.06 8.64 6.42 7.35 6.64 7.33 -Use Minitab or Excel to conduct a two-sample t-test to test the hypotheses: H₀: $\mu _ { 1 } = \mu _ { 2 }$ H₁: $\mu _ { 1 } \neq \mu _ { 2 }$ at the 5% significance level.

Portfolio An analyst is looking at two portfolios of common stocks in terms of the average price-earnings ratio.He wishes to determine if there is a difference between the two portfolios. Portfolio 1 Portfolio 2 =53 =50 =16.60 =17.62 =62.74 =127.95 -Determine the equality of the two population means.

Weight loss A new diet program claims that clients will lose weight after 7 days on the program.To validate their claim,they recorded the starting and ending weight of 11 individuals over a 7-day period.The results are shown in the following table. Starting Waight Ending Weight 221 218 215 210 206 204 236 241 214 212 256 250 240 245 222 218 239 230 218 215 229 220 -Use $\alpha$ = 0.05.State the decision rule in terms of the critical value.

VCR's An advertising executive suspects that there is a higher proportion of households which own VCR's residing in places with cable systems than there are for households residing in places without cable systems.The basis of this suspicion is the broader offering of premium programming available for taping in places with a cable system.The executive commissions a survey,which results in random samples of 165 households located in a cable system and 150 households,located outside any cable system.Of the households in the cable system,54 own VCR's; outside the cable system,33 households own VCR's.Let $\pi _ { 1 }$ be the proportion of households owning a VCR in a cable system and $\pi _ { 2 }$ be the proportion of households owning a VCR,but not in a cable system.The executive chooses 0.05 as the significance level.Can the executive reasonably conclude that the ownership of VCR's is higher in cable systems? -What is the decision rule?

The difference in the average profit margins of two industries is to be estimated for a client of a financial advisory service.Random samples from Industry A profit margins and Industry B profit margins are summarized as follows: Industry Industry =32 =33 =7.63 =6.13 =4.90 =17.68 Develop a 96-percent confidence interval for the true difference in the profit margin means for Industry A and Industry B. ____________________ to ____________________