91Ó°ÊÓ

Briefly explain the meaning of independent and dependent samples. Give one example of each.

The management at New Century Bank claims that the mean waiting time for all customers at its branches is less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample of 200 customers from the New Century Bank and found that they waited an average of \(4.5\) minutes before being served. Another sample of 300 customers taken from the Public Bank showed that these customers waited an average of \(4.75\) minutes before being served. Assume that the standard deviations for the two populations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between the two population means. b. Test at the \(2.5 \%\) significance level whether the claim of the management of the New Century Bank is true. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.01 ?\) What if \(\alpha=.05 ?\)

The following information was obtained from two independent samples selected from two populations with unknown but equal standard deviations. $$ \begin{array}{lll} n_{1}=55 & \bar{x}_{1}=90.40 & s_{1}=11.60 \\ n_{2}=50 & \bar{x}_{2}=86.30 & s_{2}=10.25 \end{array} $$ a. What is the point estimate of \(\mu_{1}-\mu_{2}\) ? b. Construct a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\).

A town that recently started a single-stream recycling program provided 60-gallon recycling bins to 25 randomly selected households and 75-gallon recycling bins to 22 randomly selected households. The total volume of recycling over a 10-week period was measured for each of the households. The average total volumes were 382 and 415 gallons for the households with the 60 - and 75 -gallon bins, respectively. The sample standard deviations were \(52.5\) and \(43.8\) gallons, respectively. Assume that the 10 -week total volumes of recycling are approximately normally distributed for both groups and that the population standard deviations are equal. a. Construct a \(98 \%\) confidence interval for the difference in the mean volumes of 10 -week recycling for the households with the 60 - and 75 -gallon bins. b. Using the \(2 \%\) significance level, can you conclude that the average 10 -week recycling volume of all households having 60 -gallon containers is different from the average volume of all households that have 75 -gallon containers?

A high school counselor wanted to know if tenth-graders at her high school tend to have more free time than the twelfth-graders. She took random samples of 25 tenth-graders and 23 twelfth-graders. Each student was asked to record the amount of free time he or she had in a typical week. The mean for the tenthgraders was found to be 29 hours of free time per week with a standard deviation of \(7.0\) hours. For the twelfthgraders, the mean was 22 hours of free time per week with a standard deviation of \(6.2\) hours. Assume that the two populations are normally distributed with equal but unknown population standard deviations. a. Make a \(90 \%\) confidence interval for the difference between the corresponding population means. b. Test at the \(5 \%\) significance level whether the two population means are different.

A random sample of nine students was selected to test for the effectiveness of a special course designed to improve memory. The following table gives the scores in a memory test given to these students before and after this course. $$ \begin{array}{l|lllllllll} \hline \text { Before } & 43 & 57 & 48 & 65 & 81 & 49 & 38 & 69 & 58 \\ \hline \text { After } & 49 & 56 & 55 & 77 & 89 & 57 & 36 & 64 & 69 \\ \hline \end{array} $$ a. Construct a \(95 \%\) confidence interval for the mean \(\mu_{d}\) of the population paired differences, where a paired difference is defined as the difference between the memory test scores of a student before and after attending this course. b. Test at the \(1 \%\) significance level whether this course makes any statistically significant improvement in the memory of all students.