Exam 21: Comparing Means

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 20 and 30 for Group 1 and Group 2,respectively.If we wish to obtain a margin of error of at most 5,what sample size should we take from each group? Assume equal sample sizes.

Compare the technique for decision making about populations using the hypothesis test method and the confidence interval method.

When testing the difference between the means of a treatment group and a placebo group,the computer display below is obtained.Using a 0.05 significance level,is there sufficient evidence to support the claim that the treatment group (variable 1)represents a population whose mean is different from 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

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 20 and 30 for Group 1 and Group 2,respectively.If we wish to obtain a margin of error of at most 10,what sample size should we take from each group? Assume equal sample sizes.

A paint manufacturer wishes to compare the drying times of two different types of paint.Independent random samples of 11 cans of type A and 9 cans of type B were selected and applied to similar surfaces.The drying times (in minutes)were recorded.The summary statistics are as follows. Type A Type B =76.1 =66.5 =4.5 =5.1 =11 =9 Do the data provide sufficient evidence to conclude that the mean drying time for type A differs from the mean drying time for type B? Perform a $\text { t-test }$ at the 1% significance level.

Researchers wanted to compare the effectiveness of a water softener used with a filtering process with a water softener used without filtering.Ninety locations were randomly divided into two groups of equal size.Group A locations used a water softener and the filtering process,while group B used only the water softener.At the end of three months,the water was tested at each location for its level of softness.(Water softness was measured on a scale of 1 to 5,with 5 being the softest water.)The results were as follows. $\text { Group A (water softener and filtering) }$ mean = 2.1 standard deviation = 0.7 $\text { Group B (water softener only) }$ mean = 1.7 standard deviation = 0.4 Refer to the sample data to test whether there is a difference between the two types of treatments.Carry out the test with a 1% significance level.

A researcher was interested in comparing the amount of time spent watching television by women and by men.Independent random samples of 14 women and 17 men were selected and each person was asked how many hours he or she had watched television during the previous week.The summary statistics are as follows. Sample 1 (women) Sample 2 (men) =12.4 =13.6 =3.9 =5.2 =14 =17 Do the data provide sufficient evidence to conclude that the mean time for women is less than the mean time for men? Perform a t-test at the 5% 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 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.Carry out the test assuming the population variances are equal.

The Better Cookie Company claims its chocolate chip cookies have more chips than another chocolate chip cookie.120 Better Cookies and 100 of the other type of cookie were randomly selected and the number of chips in each cookie was recorded.The results are as follows. Better Another Mean number of chips 7.6 6.9 Standard deviation 1.4 1.7 At the 2% level of significance,test the claim that the population of Better Cookies has a higher mean number of chips.

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 df = 52 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.