Confidence Intervals (144 problems)

Brand x and brand y of pain relievers were tested to see how much ibuprofen each table contained. Does brand x have more ibuprofen? Test at the .05 level.

In the Mediterranean sea, 600 brills were captured and dissected and 216 were found to be infected by a parasite. In the Atlantic Ocean, 120 brills were captured and dissected and 24 were found to be infected.

(a) Construct a 90% confidence interval for comparing the proportions of infected brills in the two captured sites

(b) Test whether the proportion of infected brills at the Mediterranean sea is higher that at the Atlantic Ocean

A sample of size n = 14 is selected from a normal population to construct a 95% confidence interval estimate for a population mean. The interval was computed to be (7.82 to 9.64). Determine the sample standard deviation.

Find the margin of error, E, given that the college students annual earnings has a mean, p = $3967, and sample size n = 74 and a standard deviation, a = $874 and the degree of confidence is 95%.

A survey of users of the Internet found that makes outnumbered females by nearly 2 to 1. This was a surprise, because earlier surveys had put the ratio of men to women closer to 9 to 1. Later in the article we find this information

Detailed surveys were sent to more than 13,000 organizations on the internet; 1,468 usable responses were received. According to Mr. Quarterman, the margin of error is 2.8 percent, with confidence level of 95%.

(a) What is the response rate for this survey?

(b) Do you think that the small margin of error is a good measure of the accuracy of the survey results?

Problem: A Gallup poll of 1002 adults found that 25% of the responded said that their household had experienced a crime in the past year. Among respondents aged 18 to 29 years, 43% had been victims of a crime. Gallup says, “For results based on the total sample of national adults, one can say with 95% confidence that a margin of sampling error is”. Is this margin of error for adults aged 18 to 29 smaller or larger than? Why?

a) In a study of sudden infant death syndrome [SIDS], the authors report that "7 of the 53 infants who had broncheo-pulmonary displasia [BPD] (13%) and two of the 65 control infants were victims of SIDS (p=.0392)", using the standard (uncorrected) chi-square test. Confirm this result.

b) Give a 2-tailed Fisher's exact p-value for the above comparison. What is your conclusion using this test statistic?

c) What is the RR of SIDS for an infant with BPD relative to a control infant? Give a 95% confidence interval for this RR. Verbally interpret this result.

d) What is the OR of SIDS for an infant with BPD relative to a control infant? Give a 95% confidence interval for this OR. Verbally interpret this result.

e) Compare the RR and OR derived in 3 & 4 above. Comment on this comparison.

a) In a study of sudden infant death syndrome [SIDS], the authors report that "7 of the 53 infants who had broncheo-pulmonary displasia [BPD] (13%) and two of the 65 control infants were victims of SIDS (p=.0392)", using the standard (uncorrected) chi-square test. Confirm this result.

b) Give a 2-tailed Fisher's exact p-value for the above comparison. What is your conclusion using this test statistic?

c) What is the RR of SIDS for an infant with BPD relative to a control infant? Give a 95% confidence interval for this RR. Verbally interpret this result.

d) What is the OR of SIDS for an infant with BPD relative to a control infant? Give a 95% confidence interval for this OR. Verbally interpret this result.

e) Compare the RR and OR derived in 3 & 4 above. Comment on this comparison.

In the Box population experiment this year, the nominal 90% confidence interval using the simple random sample was independently constructed 27 times. Of these intervals, 24 included the true population mean. Let p be the true probability that this interval, based on a SRS and properly constructed, includes the true mean m.

a. Ignoring any concerns you may have with the Normal approximation, compute and interpret the approximate (Wald) 95% confidence interval for p.

b. Repeat problem (a) using the Agresti-Coull interval.

c. Obtain an exact 95% confidence interval for p, using Minitab’s one-sample proportion item under Stat Þ Basic Stat.

A company wants to market a new product to their present customers. They survey 100 potential customers and find that 21 would buy the new product. If you presently have 38,232 customers, determine the actual number of customers that would purchase the new product with a 95% level of confidence.

Manager claims average sales for her shop is $1800 a day during winter months. 10 winter days selected at random, and the mean of the sales is $1830. The standard deviation of the population is $200. Can one reject the claim at a significance level of .05? Find the 95% confidence interval of the mean. Does the confidence interval interpretation agree with the hypothesis test results? Explain. Assume the variable is normally distributed.

A study was conducted to estimate the mean amount spent on birthday gifts for a typical family having two children. A sample of 150 was taken, and the mean amount spent was $225. Assuming a standard deviation equal to $50, find the 95% confidence interval for m, the mean for all such families.

A sample of 100 bean cans showed an average weight of 13 ounces with a standard deviation of 0.8 ounces. Construct a 90% confidence interval for the mean of the population.

A machine that stuffs a cheese-filled snack product can be adjusted for the amount of cheese injected into each unit. A simple random sample of 30 units is selected, and the average amount of cheese injected is found to be x = 3.5 grams. If the process standard deviation is known to be ơ = 0.25 grams, construct the 95% confidence interval for µ = the average amount of cheese being injected by the machine.

The national association of Independent colleges and universities reported that students graduating from public colleges and universities carry an average debt of $12,000 upon graduation. Assume that this average debt is based on sample of 245 students and that, based on past studies, the population standard deviation for the dept upon graduation is $2200.

a. Develop a 90% confidence interval estimate of the population mean.

b. Develop a 95% confidence interval estimate of the population mean.

c. Develop a 99% confidence interval estimate of the population mean.

d. Discuss what happens to the width of the confidence interval as the confidence level is increased. Does this result seem reasonable? Explain.

The confidence interval 75.63 < p < 77.97 is found by using the sample data for which n = 35 and X =76.8 and a = 4.2. Find the degree of confidence.

When constructing the 95% and 99% confidence intervals around a mean from some sample, two z-scores are important. What are they?

Beechcraft, Inc., wants to estimate the average time it takes the Beechjet corporate jet to climb from sea level to 41,000 feet. From previous experience, company engineers believe that the standard deviation of climbing times is two minutes. The model is tested in 100 random trials, and the sample mean is found to be 22 minutes. Give a 95% confidence interval for the average climbing time from sea level to 41,000 feet.

A park ranger has timed the mating calls of 22 bull elk. The mean duration of these calls is 14.6 seconds with standard deviation 2.8 seconds. Find a 95% confidence interval for the population mean duration of mating calls of the bull elk.

A sample of 16 students is taken. The average age in the sample was 22 years with a standard deviation of 6 years. Construct a 95 % confidence interval for the average age of the population.