Exam 11: Hypothesis Tests Involving Two Sample Means

Population In an effort to test the hypothesis that the proportion of home ownership in the state of Delaware is different than the national proportion,the following data was collected. Population Number of Home Owners Sample Size Delaware 310 500 Nation 390 600 -Use $\alpha$ = 0.01.State your conclusions in terms of the test statistic.

Golf ball A new golf ball has been designed to improve driving distance off the tee when compared to their market competitor.To test their new ball,nine golfers hit a drive with the current ball and then hit another drive with the new ball.The distance in yards for each golfer is shown in the following table. Current Golf Ball (Pupulation 1) New Golf Ball (Population 2) 212 215 236 228 250 256 243 264 250 248 235 255 230 239 225 218 220 239 -Use $\alpha$ = 0.05.State the null and alternative hypothesis.

Fast Food The general manager of a chain of fast food chicken restaurants wants to determine how effective their promotional campaigns are.In these campaigns "20% off" coupons are widely distributed.These coupons are only valid for one week.To examine their effectiveness,the executive records the daily gross sales (in $1,000s)in one restaurant during the campaign and during the week after the campaign ends.The data is shown below. Day Sales During Campaign Sales After Campaign Sunday 18.1 16.6 Monday 10.0 8.8 Tuesday 9.1 8.6 Wednesday 8.4 8.3 Thursday 10.8 10.1 Friday 13.1 12.3 Saturday 20.8 18.9 -Can they infer at the 5% significance level that sales increase during the campaign? H₀: ____________________ H₁: ____________________ Test statistic = ____________________ Conclusion: ________________________________________

Coffee breaks Do government employees take longer coffee breaks than private sector workers? That is a question that interested a management consultant.To examine the issue,he took a random sample of ten government employees and another random sample of ten private sector workers and measured the amount of time (in minutes)they spent in coffee breaks during the day.The results are listed below.Assume that the two populations are normally distributed. Government Employees Private Sector Workers 23 25 18 19 34 18 31 22 28 28 33 25 25 21 27 21 32 20 21 16 -Estimate with 95% confidence the difference in coffee breaks mean time between the two groups. ____________________ to ____________________

Realtor A new realtor in a large community is attempting to determine differences in the selling prices of houses in two sections of the community.Population 1 is the Northeast and Population 2 is the Southwest.The realtor is going to perform the hypothesis test: H₀: $\mu _ { 1 } = \mu _ { 2 }$ H₁: $\mu _ { 1 } \neq \mu _ { 2 }$ A random sample of 35 sale homes is taken from the Northeast and 41 from those in the Southwest using the multiple listing services.A Minitab summary of the results of the two samples expressed in thousands follows: N MEAN MEDIAN TRMEAN STDEV SE MEAN NORTHEAST 35 86.789 84.800 88.313 60.067 10.153 SOUTHWEST 41 60.490 62.200 61.857 14.740 2.302 -What is the value of the test statistic? z = ____________________

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.How many degrees of freedom are associated with this test?

Automotive parts A regional automotive parts chain store firm wants to improve the sales of tune-up supplies.It believes that a TV ad with a popular local,but offbeat,know-it-all actor might be able to affect their sales.Before the ads are run on TV,the company randomly samples eight of its weekly sales from past years.Following the ad campaign,eight weeks of sales were sampled.Weekly sales are approximately normally distributed and the population standard deviations are equal.Their hypothesis test is: Did the TV ad campaign help or hurt sales? To answer this question,the company plans to use a 0.05 significance level.The results of the two samples are: Before Ad Campaign After Ad Campaign =8 =8 =184 =189 =548 =456 -What is the decision rule?

Unlike the t distribution,the exact shape of the F distribution is determined by two different degrees of freedom instead of just a single value.

Real pop A marketing research firm is conducting a survey to determine if there is a difference in consumer choice in the selection of two rival soft drinks.Those surveyed were asked to rank the taste from 1-10 with 10 being the highest rating.A summary of the survey follows: Real Cola New Pep =30 =36 =7.5 =5.4 =10.6 =14.9 -Calculate the test statistic. Appropriate test: ____________________ Test statistic = ____________________

Traffic Coordinator A traffic coordinator for a community wishes to determine whether or not there is any difference in the average age of female and male drivers involved in rear-end accidents.The hypothesis test is: H₀: $\mu _ { 1 } = \mu _ { 2 }$ (no difference in mean age) H₁: $\mu _ { 1 } \neq \mu _ { 2 }$ (mean age is different)A random sample of males and females involved in rear-end accidents results in the following Minitab summary: N MEAN MEDIAN TRMEAN STDEV SEMEAN FEMALES 33 23.91 20.00 23.38 9.77 1.70 MALES 38 28.87 28.50 28.44 9.67 1.57 -What is the decision rule at the 0.025 level of significance?

Fast Food The general manager of a chain of fast food chicken restaurants wants to determine how effective their promotional campaigns are.In these campaigns "20% off" coupons are widely distributed.These coupons are only valid for one week.To examine their effectiveness,the executive records the daily gross sales (in $1,000s)in one restaurant during the campaign and during the week after the campaign ends.The data is shown below. Day Sales During Campaign Sales After Campaign Sunday 18.1 16.6 Monday 10.0 8.8 Tuesday 9.1 8.6 Wednesday 8.4 8.3 Thursday 10.8 10.1 Friday 13.1 12.3 Saturday 20.8 18.9 -Estimate with 95% confidence the mean difference.

Random samples from two normal populations produced the following statistics: n₁ = 16,s₁² = 55,n₂ = 14,s₂² = 118.Is there enough evidence at the 10% significance level to infer that the two population variances differ? Hypotheses: ____________________ Test statistic = ____________________ Conclusion: ____________________

Diet and health have become big business,but many people are not following good health practices.In 1980,a survey of 150 indicated that 30 percent said that at least one time during that year they had five or more alcoholic drinks in one sitting.In 2000,a similar survey of 150 showed that 38 percent had five drinks in one sitting at least one time during the year.Does this represent a significant increase? Use 0.01 level of significance.

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 -In testing the hypotheses: H₀: $\mu _ { 1 } - \mu _ { 2 } = 0$ H1: $\mu _ { 1 } - \mu _ { 2 } \neq 0$ t wo random samples from two normal populations produced the following statistics: n₁ = 50, $\overline { \mathrm { X } } _ { 1 }$ = 35,s₁ = 28,n₂ = 50, $\overline { \mathrm { X } } _ { 2 }$ = 30,s₂ = 10.What conclusion can we draw at the 5% significance level? Appropriate test: ____________________ Test statistic = ____________________ Conclusion: ____________________________

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 conclusion?

In testing whether the means of two normal populations are equal,summary statistics computed for two independent samples are as follows: Brand Brand =25 =25 =7.30 =6.80 =1.05 =1.20 Assume that the population variances are equal.Then,the standard error of the sampling distribution of the sample mean difference $\overline { \mathrm { x } } _ { 1 } - \overline { \mathrm { x } } _ { 2 }$ is equal to:

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 conclusion?

Smoking In an effort to test the hypothesis that the proportion of males who smoke cigarettes regularly is greater than the proportion of females who smoke cigarettes regularly,the following data was collected. Mopulation Number Wha Smoke Sample Sire Male Female 180 775 -Use $\alpha$ = 0.05.State your conclusions in terms of 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.State the null and alternative hypothesis.

Coffee breaks Do government employees take longer coffee breaks than private sector workers? That is a question that interested a management consultant.To examine the issue,he took a random sample of ten government employees and another random sample of ten private sector workers and measured the amount of time (in minutes)they spent in coffee breaks during the day.The results are listed below.Assume that the two populations are normally distributed. Government Employees Private Sector Workers 23 25 18 19 34 18 31 22 28 28 33 25 25 21 27 21 32 20 21 16 -Do these data provide sufficient evidence at the 5% significance level to support the consultant's claim? H₀: ____________________ H₁: ____________________ Appropriate test: ____________________ Test statistic = ____________________ Conclusion: ____________________ Interpretation: _________________________________________________