- A random sample of size n = 16 from a normal population yield a sample mean of x ̄ = 53.7 and standard deviation s = 23.3.
(a) Construct and interpret a 95% confidence interval for mean μ.
(b) Suppose that σ = 20. What sample size is required that our estimate is within 2 units from the mean.
2. A random sample of 15 shearing pins is taken in a study of the Rockwell hardness of the pin head. Measurements on the Rockwell hardness are made for each of the 15, yielding an average value of 47.50 with a sample standard deviation of 2.5. Assuming the measurements to be normally distributed, (a) construct a 90% confidence interval for the mean Rockwell hardness, and (b) construct a 95% prediction interval.
3.The following data represent the length of time, in days, to recovery for patients randomly treated with one of two medications to clear up severe bladder infections:
Medication 1 n1 = 14 x ̄1 = 17.4 s21 = 1.78
Medication 2 n2 = 16 x ̄2 = 19.2 s2 = 1.65
Find a 95% confidence interval for the difference μ2 − μ1 in the mean recovery times for the two medications, assuming normal populations with equal variances.
- A study is to be made to estimate the proportion of residents of a certain city and its suburbs who favor the construction of a nuclear power plant near the city.
- (a) How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 0.04 of the true proportion of residents who favor the construction of the nuclear power plant?
- (b) In a sample of 285 residents it is found that 121 are in favor the construction of the power plant. Construct a 95% confidence interval for the proportion of residents that favor the plant and give its interpretation.
- Ten engineering schools in the United States were surveyed. The sample contained 250 electrical engineers, 80 being women; 175 chemical engineers, 40 being women. Compute a 90% confidence interval for the difference between the proportions of women in these two fields of engineering. Is there a significant difference between the two proportions?
6. A random sample of size n = 64 from a Poisson random variable with unknown mean μ yields a sample mean of x ̄ = 8.5. Use the central limit theorem to find a 95% confidence interval for μ.
