1. Problem 1: The time required by workers to complete a certain job is normally distributed with a mean of 50 minutes and a standard deviation of 8 minutes.
a) What is the probability that a selected worker will require exactly 50 minutes to complete the job?
b) What is the probability that a selected worker will require less than 45 minutes to complete the job?
c) What is the 95th percentile of the required time to complete the job?
d) If the times required of sample of 25 security guards to complete the rounds is recorded, find the probability that the sample mean will be more than 20 minutes.
2. Problem 2: A medical researcher obtains the systolic blood pressure readings (in mm Hg) in the accompanying list from a sample of women aged 18-24 who have a new strain of viral infection. (Healthy women in that age group have a mean of 114.8 and a standard deviation of 13.1.)
134.9 78.7 108.9 133 123.7 96.1 126.9 89.9
132 134.7 132.1 121.7 112.3 150.2 158.3 154.4
a- Find the sample mean and standard deviation s.
b- Use a 0.05 significance level to test the claim that the sample does not come from a healthy women population.
c- Use the sample data to construct a 95% confidence interval for the population mean. Do the confidence interval limits contain the value of 114.8, which is the mean for healthy women aged 18-24?
d- Based on the preceding results, does it seem that the new strain of viral infection affects systolic blood pressure?
3. Problem 3: The following data are taken from a study investigating the length of time in days between the occurrence of the injury and the first magnetic resonance imaging (MRI). The data are shown in the following table:
Subject Days Subject Days Subject Days Subject Days
1 14 6 0 11 28 16 14
2 9 7 10 12 24 17 9
3 18 8 4 13 24
4 26 9 8 14 2
5 12 10 21 15 3
a- Compute a point estimate for mean number of days between injury and initial MRI and standard error of the mean.
b- Test whether the mean number of days between injury and initial MRI is not 14 days in a population presumed to be represented by these sample data?
c- Interpret your answer in a.
d- Find the 95% confidence interval of the mean number of days between injury and initial MRI?
e- Explain what does “a 95% confidence” means
4. Problem 4: As part of the National Health Survey conducted by the Department of Health and Human Services, self-reported heights and measured heights were obtained for females aged 12-16. Listed below are sample results.
Reported height 53 64 61 64 66 65 68 63 64 64 64 67
Measured height 58.1 62.7 61.1 63 64.8 66.4 67.6 63.5 67 63.5 62.1 68
a- Is there sufficient evidence to support the claim that there is a difference between self-reported heights and measured heights of females aged 12-16? Use a 0.05 significance level.
b- Construct a 95% confidence interval estimate of the mean difference between reported heights and measured heights. Interpret the resulting confidence interval.
5. Problem 5: People spend huge sums of money (currently around $5 billion annually) for the purchase of magnets used to treat a wide variety of pains. Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results given below are among the results obtained in the study (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). Assume that the distributions of reduction in pain level after magnet treatment and sham treatment are normal.
Reduction in pain level after magnet treatment: n=20, Sample mean=0.49, s = 0.96
Reduction in pain level after sham treatment: n=20, Sample mean=0.44, s=1.4
a- Use a 0.05 significance level to test the claim that those treated with magnets have a greater reduction in pain than those given a sham treatment.
b- Construct a 95% confidence interval estimate of the difference between the mean reduction in pain for those treated with magnets and the mean reduction in pain for those given a sham treatment.