CONFIDENCE INTERVAL PROBLEMS
- Watch your cholesterol: A sample of 314 patients between the ages of 38 and 82 were given a combination of the drugs ezetimibe and simvastatin. They achieved a mean reduction in total cholesterol of 0.94 millimole per liter. Assume the sample standard deviation is 0.18.
a)Construct a 98% confidence interval for the mean reduction in total cholesterol in patients who take this combination of drugs.
b)Should this confidence interval be used to estimate the mean reduction in total cholesterol for patients over the age of 85? Explain.
c)Based on the confidence interval constructed in part (a), is it likely that the mean reduction in cholesterol level is less than 0.90?
[Navidi; page 303]
- SAT scores: A college admissions officer takes a simple random sample of 100 entering freshmen and computes their mean mathematics SAT score to be 458. Assume the population standard deviation is 116.
a)Construct a 99% confidence interval for the mean mathematics SAT score for the entering freshman class.
b)If the sample size were 75 rather than 100, would the margin of error be larger or smaller than the result in part (a)? Explain.
c)If the confidence level were 95% rather than 99%, would the margin of error be larger or smaller than the result in part (a)? Explain.
[Navidi; pg 303]
- Babies: According to the National Health Statistics Reports, a sample of 360 one-year-old baby boys in the United States had a mean weight of 25.5 pounds. Assume the population standard deviation is 5.3 pounds.
a)Construct a 95% confidence interval for the mean weight of all one-year-old baby boys in the United States.
b)Should this confidence interval be used to estimate the mean weight of all one-year-old babies in the United States? Explain.
c)Based on the confidence interval constructed in part (a), is it likely that the mean of all one-year-old boys is less than 28 pounds?
[Navidi; pg 303]
- Sound it out: Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. A sample of 134 first-graders who were learning English were asked to identify as many letter sounds as possible in a period of one minute. The average number of letter sounds identified was 34.06 with a standard deviation of 23.83.
a)Construct a 98% confidence interval for the mean number of letter sounds identified in one minute. (29.19, 38.93)
b)If a 95% confidence interval were constructed with these data, would it be wider or narrower than the interval constructed in part a? Explain. Narrower
[Navidi; pg 315]
- Software instruction: A hybrid course is one that contains both online and classroom instruction. In a study performed at Macon State College, a software package was used as the main source of instruction in a hybrid college algebra course. The software tracked the number of hours it took for each student to meet the objectives of the course. In a sample of 45 students, the mean number of hours was 80.5, with a standard deviation of 51.2.
a)Construct a 95% confidence interval for the mean number of hours it takes for a student to meet the course objectives.
b)If a sample of 90 students had been studied, would you expect the confidence interval to be wider or narrower than the interval constructed in part a?
[Navidi; pg 315]
- Baby talk: In a sample of 77 children, the mean age at which they first began to combine words was 16.51 months, with a standard deviation of 9.59 months.
a)Construct a 95% confidence interval for the mean age at which children first began to combine words.
b)If a sample of 50 children had been studied, would you expect the confidence interval to be wider or narrower than the interval in part a? Explain.
[Navidi; pg 315]
- Working at home: According to the U.S. Census Bureau, 43% of men who worked at home were college graduates. In a sample of 500 women who worked at home, 162 were college graduates.
a)Find a point estimate for the proportion of college graduates among women who work at home.
b)Construct a 98% confidence interval for the proportion of women who work at home who are college graduates.
c)Based on the confidence interval, is it reasonable to believe that the proportion of college graduates among women who work at home is the same as the proportion of college graduates among men who work at home? Explain.
[Navidi; pg 327]
- Sleep apnea: Sleep apnea is a disorder in which there are pauses in breathing during sleep. People with this condition must wake up frequently to breathe. In a sample of 427 people aged 65 and over, 104 of them had sleep apnea.
a)Find a point estimate for the population proportion of those aged 65 and over who have sleep apnea.
b)Construct a 99% confidence interval for the proportion of those aged 65 and over who have sleep apnea.
c)In another study, medical researchers concluded that more than 9% of elderly people have sleep apnea. Based on the confidence interval, does it appear that more than 9% of people aged 65 and over have sleep apnea? Explain.
[Navidi; page 327]
- Health insurance: In 2008, the General Social Survey asked 182 people whether they received health insurance as a benefit from their employer. A total of 60 people said they did.
a)Find a point estimate for the proportion of people who receive health insurance from their employer.
b)Construct a 95% confidence interval for the proportion of people who receive health insurance from their employer.
c)An economist states that 50% of employees receive health insurance from their employer. Does the confidence interval contradict this statement? Explain.
[Navidi; pg 327]
- WOW: In the computer game World of Warcraft, some of the strikes are critical strikes, which do more damage. Assume that the probability of a critical strike is the same for every attack, and that the attacks are independent. During a particular fight, a character has 242 critical strikes out of 595 attacks.
a)Construct a 95% confidence interval for the proportion of strikes that are critical strikes.
b)Construct a 98% confidence interval for the proportion of strikes that are critical strikes.
c)What is the effect of increasing the level of confidence on the width of the interval?
[Navidi; pg 327]
- Contaminated water: In a sample of 42 water specimens taken from a construction site, 26 contained detectable levels of lead.
a)Construct a 90% confidence interval for the proportion of water specimens that contain detectable levels of lead.
b)Construct a 95% confidence interval for the proportion of water specimens that contain detectable levels of lead.
c)What is the effect of increasing the level of confidence on the width of the interval?
[Navidi; pg 327]
- Internet service: An Internet service provider sampled 540 customers, and finds that 75 of them experienced an interruption in high-speed service during the previous month.
a)Find a point estimate for the population proportion of all customers who experienced an interruption.
b)Construct a 90% confidence interval for the proportion of all customers who experienced an interruption.
c)The company’s quality control manager claims that no more than 10% of its customers experienced an interruption during the previous month. Does the confidence interval contradict this claim? Explain.
[Navidi; pg 327]