Exam 21: Comparing Means

A researcher is interested in the academic performance differences between individuals using an optimistic versus a pessimistic approach to their studies.If the researcher fails to find a significant difference,when in fact one exists in the population,

When testing the difference between the means of a treatment group and a placebo group,the computer display below is obtained.Using a 0.01 significance level,is there sufficient evidence to support the claim that the treatment group (variable 1)represents a population whose mean is greater than the mean for the untreated population? Explain. t-Test: Two Sample for Means 1 Variable 1 Variable 2 2 Mean 171.6392 168.7718 3 Known Variance 47.51672 41.08293 4 Observations 50 50 5 Hypothesized Mean Difference 0 6 t 2.154057 7 P(T>=t) one-tail 0.0158 8 TCritical one-tail 1.644853 9 P(T>=t) two-tail 0.0316 10 tCritical two-tail 1.959961

A researcher was interested in comparing the GPAs of students at the University of Toronto Scarborough and Mississauga campuses.Independent random samples of 8 students from the Scarborough campus and 13 students from the Mississauga campus yielded the following GPAs. Scarborough Mississauga 3.7 3.8 2.8 3.2 3.2 4.0 3.0 3.0 3.6 2.5 3.9 2.6 2.7 3.8 4.0 3.6 2.5 3.6 2.8 3.9 3.4 Determine a 95% confidence interval for the difference, $\mu _ { 1 } - \mu _ { 2 } ,$ between the mean GPA of all students at the Scarborough campus and the mean GPA of all students at the Mississauga campus.

You wish to construct a 90% confidence interval to compare the mean measurement for two groups.A small pilot study yields sample standard deviations of 10 and 15 for Group 1 and Group 2,respectively.If we wish to obtain a margin of error of at most 3,what sample size should we take from each group? Assume equal sample sizes.

The table below gives information concerning the gasoline mileage for random samples of trucks of two different types.Find a 95% confidence interval for the difference in the means $\mu_\mathrm { X }$ - $\mu_Y$ . Brand Brand Y Number of Trucks 50 50 Mean mileage 20.4 24.9 Standard Deviation 2.3 1.8

You wish to construct a 95% confidence interval to compare the mean measurement for two groups.A small pilot study yields sample standard deviations of 10 and 12 for Group 1 and Group 2,respectively.If we wish to obtain a margin of error of at most 2,what sample size should we take from each group? Assume equal sample sizes.

A researcher was interested in comparing the salaries of female and male employees of a particular company.Independent random samples of 8 female employees (sample 1)and 15 male employees (sample 2)yielded the following weekly salaries (in dollars). Female Male 495 722 518 760 562 904 556 880 1150 904 520 805 520 500 480 1005 1250 970 743 750 605 660 1640 Determine a 98% confidence interval for the difference, $\mu _ { 1 } - \mu _ { 2 } ,$ between the mean weekly salary of all female employees and the mean weekly salary of all male employees.

A grocery store manager is interested in determining whether or not a difference exists between the shelf life of two different brands of doughnuts.A random sample of 100 boxes of each brand was selected and the shelf life in days was determined for each box.The sample results are given below. Brand A Brand B =2.1 =2.9 =0.8 =1.1 =100 =100 Find a 98% confidence interval for $\mu _ { \mathrm { A } }$ - $\mu_\mathrm { B }$ ,that is,the difference in mean shelf life between Brand A and Brand B.

A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet.Subjects were randomly assigned to a treatment group and a control group.The mean blood pressure was determined for each group,and a 90% confidence interval for the difference in the mean between the treatment group and the control group, $\mu _ { \mathrm { t } } - \mu _ { \mathrm { C } }$ was found to be (-26,-6).

The data below show the number of tornadoes spotted in a midwestern state in the U.S.A.annually before and after 1960.A researcher wants to determine whether there has been a change in the frequency of tornadoes in this state. 1940-1959 1960-1979 2,2,2,9,4, 4,0,1,2,2 2,0,1,2,1, 1,3,2,4,5 2,3,3,5,4, 1,2,4,2,1 5,2,4,0,4 0,2,4,2,2

A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet.Use the sample data below to construct a 99% confidence interval for $\mu _ { 1 }$ - $\mu 2$ ,where $\mu _ { 1 }$ and $\mu 2$ represent the population means for the treatment group and the control group,respectively. $\text { Treatment Group }$ $\text { Control Group }$ n1 = 85 n2 = 75 $\bar { x } _ { 1 }$ = 189.1 $\bar { x } _ { 2 }$ = 203.7 s1 = 38.7 s2 = 39.2

You wish to construct a 90% confidence interval to compare the mean measurement for two groups.A small pilot study yields sample standard deviations of 10 and 12 for Group 1 and Group 2,respectively.If we wish to obtain a margin of error of at most 2,what sample size should we take from each group? Assume equal sample sizes.

A study was made to determine which taxi company gave quicker service.Companies A and B were each called at 50 randomly selected times.The response times were recorded.The results are as follows. Company A Company B Mean response time 7.6 minutes 6.9 minutes Standard deviation 1.4 minutes 1.7 minutes At the 0.02 level of significance,test the claim that the two companies have different mean response times.Carry out the test assuming the population variances are equal.

When testing for a difference between the means of a treated population and an untreated population,the computer display below is obtained.Explain what the P-Value of 0.0316 means in this context. t-Test: Two Sample for Means 1 Variable 1 Variable 2 2 Mean 171.6392 168.7718 3 Known Variance 47.51672 41.08293 4 Observations 50 50 5 Hypothesized Mean Difference 0 6 t 2.154057 7 P(T>=t) one-tail 0.0158 8 TCritical one-tail 1.644853 9 P(T>=t) two-tail 0.0316 10 tCritical two-tail 1.959961

A study was made to determine which taxi company gave quicker service.Companies A and B were each called at 50 randomly selected times.The response times were recorded.The results are as follows. Company A Company B Mean response time 7.6 minutes 6.9 minutes Standard deviation 1.4 minutes 1.7 minutes df = 94. At the 0.02 level of significance,test the claim that the two companies have different mean response times.

A grocery store is interested in determining whether or not a difference exists between the shelf life of Tasty Choice doughnuts and Sugar Twist doughnuts.A random sample of 100 boxes of each brand was selected and the mean shelf life in days was determined for each brand.A 98% confidence interval for the difference of the means, $\mu \mathrm { TC }$ - $\mu \mathrm { ST }$ ,was determined to be (1.1,2.8).

A researcher was interested in comparing the GPAs of students at the University of Toronto Scarborough and Mississauga campuses.Independent random samples of 8 students from the Scarborough campus and 13 students from the Mississauga campus yielded the following GPAs. Scarborough Mississauga 3.7 3.8 2.8 3.2 3.2 4.0 3.0 3.0 3.6 2.5 3.9 2.6 2.7 3.8 4.0 3.6 2.5 3.6 2.8 3.9 3.4 Do the data provide sufficient evidence to conclude that the mean GPA of students at the Scarborough campus differs from the mean GPA of students at the Mississauga campus? Perform a t-test at the 10% significance level.

A researcher was interested in comparing the salaries of female and male employees of a particular company.Independent random samples of female employees (sample 1)and male employees (sample 2)were taken to calculate the mean salary,in dollars per week,for each group.A 98% confidence interval for the difference, $\mu _ { 1 } - \mu _ { 2 } ,$ between the mean weekly salary of all female employees and the mean weekly salary of all male employees was determined to be $( - \$ 180 , \$ 40 )$

Two types of flares were tested for their burning times (in minutes).Sample results are given below. $\text { Brand } \mathrm { X }$ $\text { Brand } \mathrm { Y }$ n = 35 n = 40 $\overline{x}$ = 19.4 $\overline{x}$ = 15.1 s = 1.4 s = 0.8 Refer to the sample data to test the claim that the two populations have unequal means.Carry out the test with a 1% significance level.

Two machines are being considered to weigh luggage at an airport.A standard 30 kg weight is used to test the machines.Sample weigh-ins were taken from each machine and the following results were obtained. Machine 1 Machine 2 Sample size 25 36 Sample mean 31.8 30.9 Sample SD 2.2 1.98 df = 48. At the 0.02 level of significance,test the claim that the population mean for Machine 1 is different from the population mean for Machine 2.