# Statistics Problem Set: Intervals

5.2. What is the confidence level of each of the following confidence intervals for the mean? (see attached).

5.20. The "Raid" test kitchen. According to scientists, the cockroach has had 300 million years to develop a resistance to destruction. In a study conducted by researchers for S.C. Johnson & Son, Inc. (manufacturers of Raid), 5,000 roaches (the expected number in a roach-infested house) were released in the Raid test kitchen. One week later, the kitchen was fumigated, and 16,298 dead roaches were counted, a gain of 11,298% roaches for the 1-week period. Assume that none of the original roaches died during the 1-week period and that the standard deviation of the number of roaches produced per roach in a 1-week period is 1.5. Use the number of roaches produced by the sample of 5,000 roaches to find a 95% confidence interval for the mean number of roaches produced per week for each roach in a typical roach-infested house.

5.4. A random sample of 90 observations produced a mean of 25.9 and a standard deviation of 2.7.

a. Find an approximate 95% confidence interval for the population mean.

b. Find an approximate 90% confidence interval for the mean.

c. Find an approximate 99% confidence interval for the mean.

5.10. Latex allergy in health care workers. Health care workers who use latex gloves with glove powder on a daily basis are particularly susceptible to developing a latex allergy. Symptoms of a latex allergy include conjunctivitis, hand eczema, nasal congestion, skin rash, and shortness of breath. Each in a sample of 46 hospital employees who were diagnosed with latex allergy based on a skin-prick test reported their exposure to latex gloves. Summary statistics for the number of latex gloves used per week are mean =19.3 and standard deviation = 11.9.

a. Give a point estimate for the average number of latex gloves used per week by all health care workers with a latex allergy.

b. Form a 95% confidence interval for the average number of latex gloves used per week by all health care workers with a latex allergy.

c. Give a practical interpretation of the interval, part b.

d. Give the conditions required for the interval, part b, to be valid.

#### Solution Preview

5.2. a. confidence level =P(z<1.96)-P(z<-1.96)=0.975-0.025=0.95. Therefore, it is 95% confidence interval.

b. confidence level=P(z<1.645)-P(z<-1.645)=0.95-0.05=0.90. Therefore, it is 90% confidence interval.

c. confidence level=P(z<2.575)-P(z<-2.575)=0.995-0.005=0.99. Therefore, it is 99% confidence interval.

d. confidence level=P(z<1.282)-P(z<-1.282)=0.90-0.10=0.80. Therefore, it is 80% confidence interval.

e. confidence level=P(z<0.99)-P(z<-0.99)=0.84-0.16=0.68. Therefore, it is ...

#### Solution Summary

The confidence levels of confidence intervals for the mean are determined.