Exam 10: Two-Sample Confidence Intervals

Using technology, construct the confidence interval for the difference $\mu _ { 1 } - \mu _ { 2 }$ for the given level and values of $\bar { x } _ { 1 } , \bar { x } _ { 2 } , s _ { 1 } , s _ { 2 } , n _ { 1 }$ , and $n _ { 2 }$ . Level $98 \% , \bar { x } _ { 1 } = 26.4 , \bar { x } _ { 2 } = 22.9 , s _ { 1 } = 7.0 , s _ { 2 } = 5.4 , n _ { 1 } = 20$ , and $n _ { 2 } = 21$

The concentration of hexane (a common solvent) was measured in units of micrograms per liter for a simple random sample of twelve specimens of untreated ground water taken near a municipal Landfill. The sample mean was 720.2 with a sample standard deviation of 8.8. Eleven specimens of Treated ground water had an average hexane concentration of 695.1 with a standard deviation of 9.1. It is reasonable to assume that both samples come from populations that are approximately normal. Construct A 90% confidence interval for the reduction of hexane concentration after treatment.

The following MINITAB output display presents a 95% confidence interval for the difference between two means. Mean StDev SE Mean 9 55.539 23.660 7.887 7 89.948 25.488 9.634 $\text { What is the point estimate of } \mu _ { 1 } - \mu _ { 2 } ?$

The following display from a TI-84 Plus calculator presents a $95 \%$ confidence interval for the difference between two means. The sample sizes are $n _ { 1 } = 10$ and $n _ { 2 } = 15$ . How many degrees of freedom did the calculator use?

In an agricultural experiment, the effects of two fertilizers on the production of oranges were measured. Fourteen randomly selected plots of land were treated with fertilizer A, and 10 randomly Selected plots were treated with fertilizer B. The number of pounds of harvested fruit was measured From each plot. Following are the results. Assume that the populations are approximately normal. Construct a 95% confidence interval for the Difference between the mean yields for the two types of fertilizer. Based on your results, is it Reasonable to conclude that the mean yields may be the same for fertilizers A and B?

A sample of students is enrolled in an online statistics class, and another sample is enrolled in a traditional statistics class. At the end of the semester, the students are given a test. The scores from Each sample are compared to determine which class was more effective. Are these samples Independent or paired?

An amateur golfer wishes to determine if there is a difference between the drive distances of her two favorite drivers. (A driver is a specialized club for driving the golf ball down range.) She hits fourteen balls With driver A and 10 balls with driver B. The drive distances (in yards) for the trials are show below. Assume that the populations are approximately normal. Construct a 98% confidence interval for the Difference between the mean drive distances for the two drivers. Based on your results, is it Reasonable to conclude that the mean drive distances may be the same for drivers A and B? A) Yes B) No

In a random sample of 65 patients undergoing a standard surgical procedure, 12 required medication for postoperative pain. In a random sample of 90 patients undergoing a new procedure, only 14 Required medication. Construct a 98% confidence interval for the difference in the proportions of patients needing pain Medication between the old and new procedures.

A computer software magazine compares the rates of malware infection for computers protected by security software A with the rates of infection for computers protected by security software B. They found that out of 794 computers with security software A, 24 became infected with some type of Malware after 1000 hours of internet interaction. For security software B, 47 out of 522 computers Became infected after 1000 hours of internet interaction. Assuming these to be random samples of infection rates for the two security software packages, construct a 95% confidence interval for the difference between the proportions of infection for the two types of Security software packages.

Two microprocessors are compared on a sample of 6 benchmark codes to determine whether there is a difference in speed. The times (in seconds) used by each processor on each code are given below: Find the 95% confidence interval for the difference between the mean speeds.

Using technology, solve the following problem: An amateur golfer wishes to determine if there is a difference between the drive distances of her two favorite drivers. (A driver is a specialized club for Driving the golf ball down range.) She hits fourteen balls with driver A and 10 balls with driver B. The drive Distances (in yards) for the trials are show below. Assume that the populations are approximately normal. Construct a 95% confidence interval for the Difference between the mean drive distances for the two drivers.

A survey of college students reported that in a sample of 371 male students, the average number of energy drinks consumed per month was 2.47 with a standard deviation of 4.70, and in a sample of 282 female students, the average was 1.45 with a standard deviation of 3.12. Construct a 98% confidence interval for the difference between men and women in the mean number Of energy drinks consumed.

The following display from a TI-84 Plus calculator presents a 95% confidence interval for the difference between two means. The sample sizes are $n _ { 1 } = 8 \text { and } n _ { 2 } = 13$ $\text { Compute the point estimate of } \mu _ { 1 } - \mu _ { 2 } \text {. }$

A group of six individuals with high cholesterol levels were given a new diet designed to lower cholesterol levels. Cholesterol levels, in milligrams per deciliter, were measured before and after the Implementation of the diet for each individual, with the following results: Individual Before After 1 227 208 2 239 208 3 255 208 4 242 187 5 240 212 6 241 196 Find the 98% confidence interval for the mean reduction in cholesterol level.

The following output from MINITAB presents a confidence interval for the mean difference between matched pairs. Mean StDev SEMean Difference 16 4.9474 6.3070 1.57675 $95 \%$ CI for mean difference: $( 1.586646,8.308154 )$ What is the point estimate of $\mu _ { d }$ ?

Construct the confidence interval for the difference $\mu _ { 1 } - \mu _ { 2 }$ for the given level and values of $\bar { x } _ { 1 } , \bar { x } _ { 2 }$ , $s _ { 1 } , s _ { 2 } , n _ { 1 }$ , and $n _ { 2 }$ . Level $90 \% , \bar { x } _ { 1 } = 91.6 , \bar { x } _ { 2 } = 79.2 , s _ { 1 } = 7.1 , s _ { 2 } = 7.3 , n _ { 1 } = 15$ , and $n _ { 2 } = 11$

A group of six individuals with high cholesterol levels were given a new diet designed to lower cholesterol levels. Cholesterol levels, in milligrams per deciliter, were measured before and after the Implementation of the diet for each individual, with the following results: Individual Before After 1 227 160 2 234 174 3 257 178 4 269 170 5 218 178 6 248 178 A dietician claims that the mean reduction in cholesterol level is 51 milligrams per deciliter. Does the 98% confidence interval contradict this claim? (Hint: you need to find the 98% confidence interval For the mean reduction in cholesterol level.)

A computer software magazine compares the rates of malware infection for computers protected by security software A with the rates of infection for computers protected by security software B. They found that out of 787 computers with security software A, 72 became infected with some type of Malware after 1000 hours of internet interaction. For security software B, 37 out of 834 computers Became infected after 1000 hours of internet interaction. Assuming these to be random samples of infection rates for the two security software packages, construct a 98% confidence interval for the difference between the proportions of infection for the two types of Security software packages. Does the confidence interval contradict the claim that the proportion of Infections is the same for the two types of security software?

The following display from a TI-84 Plus calculator presents a $95 \%$ confidence interval for the difference between two means. The sample sizes are $n _ { 1 } = 13$ and $n _ { 2 } = 8$ . Fill in the blanks: We are 95% confident that the difference between the means is between _________ and _________.

Using technology, solve the following problem: The concentration of hexane (a common solvent) was measured in units of micrograms per liter for a simple random sample of fourteen specimens of Untreated ground water taken near a municipal landfill. The sample mean was 241.8 with a sample Standard deviation of 8.8. Ten specimens of treated ground water had an average hexane Concentration of 149.5 with a standard deviation of 7.4. It is reasonable to assume that both samples come from populations that are approximately normal. Construct A 95% confidence interval for the reduction of hexane concentration after treatment.