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Confidence Intervals: Homework

Module by: Dr. Barbara Illowsky, Susan Dean

Note:

If you are using a student-t distribution for a homework problem below, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Exercise 1

Among various ethnic groups, the standard deviation of heights is known to be approximately 3 inches. We wish to construct a 95% confidence interval for the average height of male Swedes. 48 male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches.

  • a.
    • i. x¯=x¯= size 12{ {overline {x}} ={}} {}________________
    • ii. σ=σ= size 12{σ} {} ________________
    • iii. s x = s x = size 12{s rSub { size 8{x} } ={}} {} ________________
    • iv. n=n= size 12{n} {}________________
    • v. n−1=n−1= size 12{n - 1} {}________________
  • b. Define the Random Variables XX size 12{X} {} and X¯X¯ size 12{ {overline {X}} } {}, in words.
  • c. Which distribution should you use for this problem? Explain your choice.
  • d. Construct a 95% confidence interval for the population average height of male Swedes.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • e. What will happen to the level of confidence obtained if 1000 male Swedes are surveyed instead of 48? Why?

Solution 1

  • a.
    • i. 71
    • ii. 3
    • iii. 2.8
    • iv. 48
    • v. 47
  • c. ( 71 , 3 48 ) ( 71 , 3 48 ) size 12{ \( "71", { { size 8{3} } over { size 8{ sqrt {"48"} } } } \) } {}
  • d.
    • i. CI: (70.15,71.85)
    • iii. EB = 0.85

Exercise 2

In six packages of “The Flintstones® Real Fruit Snacks” there were 5 Bam-Bam snack pieces. The total number of snack pieces in the six bags was 68. We wish to calculate a 96% confidence interval for the population proportion of Bam-Bam snack pieces.

  • a. Define the Random Variables XX size 12{X} {} and P'P' size 12{P'} {}, in words.
  • b. Which distribution should you use for this problem? Explain your choice
  • c. Calculate p'p' size 12{p'} {}.
  • d. Construct a 96% confidence interval for the population proportion of Bam-Bam snack pieces per bag.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • e. Do you think that six packages of fruit snacks yield enough data to give accurate results? Why or why not?

Exercise 3

A random survey of enrollment at 35 community colleges across the United States yielded the following figures (source: Microsoft Bookshelf): 6414; 1550; 2109; 9350; 21828; 4300; 5944; 5722; 2825; 2044; 5481; 5200; 5853; 2750; 10012; 6357; 27000; 9414; 7681; 3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493; 2771; 2861; 1263; 7285; 28165; 5080; 11622. Assume the underlying population is normal.

  • a.
    • i. x¯=x¯= size 12{ {overline {x}} ={}} {}
    • ii. s x = s x = size 12{s rSub { size 8{x} } ={}} {} ________________
    • iii. n=n= size 12{n} {} ________________
    • iv. n−1=n−1= size 12{n - 1} {}________________
  • b. Define the Random Variables XX size 12{X} {} and X¯X¯ size 12{ {overline {X}} } {}, in words.
  • c. Which distribution should you use for this problem? Explain your choice.
  • d. Construct a 95% confidence interval for the population average enrollment at community colleges in the United States.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • e. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? Why?

Solution 3

  • a.
    • i. 8629
    • ii. 6944
    • iii. 35
    • iv. 34
  • c. t 34 t 34 size 12{t rSub { size 8{"34"} } } {}
  • d.
    • i. CI: (6243, 11,014)
    • iii. EB = 2385
  • e. It will become smaller

Exercise 4

From a stack of IEEE Spectrum magazines, announcements for 84 upcoming engineering conferences were randomly picked. The average length of the conferences was 3.94 days, with a standard deviation of 1.28 days. Assume the underlying population is normal.

  • a. Define the Random Variables XX size 12{X} {} and X¯X¯ size 12{ {overline {X}} } {}, in words.
  • b. Which distribution should you use for this problem? Explain your choice.
  • c. Construct a 95% confidence interval for the population average length of engineering conferences.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.

Exercise 5

Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the average amount of time individuals waste at the courthouse waiting to be called for service. The committee randomly surveyed 81 people. The sample average was 8 hours with a sample standard deviation of 4 hours.

  • a.
    • i. x¯=x¯= size 12{ {overline {x}} ={}} {}________________
    • ii. s x = s x = size 12{s rSub { size 8{x} } ={}} {} ________________
    • iii. n=n= size 12{n} {}________________
    • iv. n−1=n−1= size 12{n - 1} {}________________
  • b. Define the Random Variables XX size 12{X} {} and X¯X¯ size 12{ {overline {X}} } {}, in words.
  • c. Which distribution should you use for this problem? Explain your choice.
  • d. Construct a 95% confidence interval for the population average time wasted.
    • a. State the confidence interval.
    • b. Sketch the graph.
    • c. Calculate the error bound.
  • e. Explain in a complete sentence what the confidence interval means.

Solution 5

  • a.
    • i. 8
    • ii. 4
    • iii. 81
    • iv. 80
  • c. t 80 t 80 size 12{t rSub { size 8{"80"} } } {}
  • d.
    • i. CI: (7.12, 8.88)
    • iii. EB = 0.88

Exercise 6

Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. It randomly surveys 100 people. The sample average is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal.

  • a.
    • i. x¯=x¯= size 12{ {overline {x}} ={}} {} ________________
    • ii. σ=σ= size 12{σ} {}________________
    • iii. s x = s x = size 12{s rSub { size 8{x} } ={}} {} ________________
    • iv. n=n= size 12{n} {} ________________
    • v. n−1=n−1= size 12{n - 1} {}________________
  • b. Define the Random Variables XX size 12{X} {} and X¯X¯ size 12{ {overline {X}} } {}, in words.
  • c. Which distribution should you use for this problem? Explain your choice.
  • d. Construct a 90% confidence interval for the population average time to complete the tax forms.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • e. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?
  • f. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why?
  • g. Suppose that the firm decided that it needed to be at least 96% confident of the population average length of time to within 1 hour. How would the number of people the firm surveys change? Why?

Exercise 7

A sample of 16 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was 2 ounces with a standard deviation of 0.12 ounces. The population standard deviation is known to be 0.1 ounce.

  • a.
    • i. x¯=x¯= size 12{ {overline {x}} ={}} {} ________________
    • ii. σ=σ= size 12{σ} {} ________________
    • iii. s x = s x = size 12{s rSub { size 8{x} } ={}} {} ________________
    • iv. n=n= size 12{n} {} ________________
    • v. n−1=n−1= size 12{n - 1} {} ________________
  • b. Define the Random Variable XX size 12{X} {}, in words.
  • c. Define the Random Variable X¯X¯ size 12{ {overline {X}} } {}, in words.
  • d. Which distribution should you use for this problem? Explain your choice.
  • e. Construct a 90% confidence interval for the population average weight of the candies.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • f. Construct a 98% confidence interval for the population average weight of the candies.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • g. In complete sentences, explain why the confidence interval in (f) is larger than the confidence interval in (e).
  • h. In complete sentences, give an interpretation of what the interval in (f) means.

Solution 7

  • a.
    • i. 2
    • ii. 0.1
    • iii . 0.12
    • iv. 16
    • v. 15
  • b. the weight of 1 small bag of candies
  • c. the average weight of 16 small bags of candies
  • d. ( 2, 0 . 1 16 ) ( 2, 0 . 1 16 ) size 12{ \( 2, { { size 8{0 "." 1} } over { size 8{ sqrt {"16"} } } } \) } {}
  • e.
    • i. CI: (1.96, 2.04)
    • iii. EB = 0.04
  • f.
    • i. CI: (1.94, 2.06)
    • iii. EB = 0.06

Exercise 8

A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of 9 patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4 .

  • a.
    • i. x¯=________x¯= size 12{ {overline {x}} ={}} {}________
    • ii. s x = ________ s x = size 12{s rSub { size 8{x} } ={}} {} ________
    • iii. n=________n= size 12{n} {}________
    • iv. n−1=________n−1= size 12{n - 1} {}________
  • b. Define the Random Variable XX size 12{X} {}, in words.
  • c. Define the Random Variable X¯X¯ size 12{ {overline {X}} } {}, in words.
  • d. Which distribution should you use for this problem? Explain your choice.
  • e. Construct a 95% confidence interval for the population average length of time.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • f. What does it mean to be “95% confident” in this problem?

Exercise 9

Suppose that 14 children were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of 6 months with a sample standard deviation of 3 months. Assume that the underlying population distribution is normal.

  • a.
    • i. x¯=________x¯= size 12{ {overline {x}} ={}} {}________
    • ii. s x = ________ s x = size 12{s rSub { size 8{x} } ={}} {} ________
    • iii. n=________n= size 12{n} {}________
    • iv. n−1=________n−1= size 12{n - 1} {}________
  • b. Define the Random Variable XX size 12{X} {}, in words.
  • c. Define the Random Variable X¯X¯ size 12{ {overline {X}} } {}, in words.
  • d. Which distribution should you use for this problem? Explain your choice.
  • e. Construct a 99% confidence interval for the population average length of time using training wheels.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • f. Why would the error bound change if the confidence level was lowered to 90%?

Solution 9

  • a.
    • i. 6
    • ii. 3
    • iii. 14
    • iv. 13
  • b. the time for a child to remove his training wheels
  • c. the average time for 14 children to remove their training wheels.
  • d. t 13 t 13 size 12{t rSub { size 8{"13"} } } {}
  • e.
    • i. CI: (3.58, 8.42)
    • iii. EB = 2.42

Exercise 10

Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.

  • a. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?
  • b. If it was later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?

Exercise 11

Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and found that 320 claimed to always buckle up. We are interested in the population proportion of drivers who claim to always buckle up.

  • a.
    • i. x=________x= size 12{x} {}________
    • ii. n=________n= size 12{n} {}________
    • iii. p'=________p'= size 12{p'} {}________
  • b. Define the Random Variables XX size 12{X} {} and P'P' size 12{P'} {}, in words.
  • c. Which distribution should you use for this problem? Explain your choice.
  • d. Construct a 95% confidence interval for the population proportion that claim to always buckle up.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • e. If this survey were done by telephone, list 3 difficulties the companies might have in obtaining random results.

Solution 11

  • a.
    • i. 320
    • ii . 400
    • iii. 0.80
  • c. N ( 0 . 80 , ( 0 . 61 ) ( 0 . 39 ) 1200 ) N ( 0 . 80 , ( 0 . 61 ) ( 0 . 39 ) 1200 ) size 12{N \( 0 "." "80", { { size 8{ \( sqrt {0 "." "61"} \) \( 0 "." "39"} } over { size 8{"1200"} } } \) } {}
  • d.
    • i. CI: (0.76, 0.84)
    • iii. EB = 0.02

Exercise 12

Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its average number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats.

  • a.
    • i. x¯=________x¯= size 12{ {overline {x}} ={}} {}________
    • ii. s x = ________ s x = size 12{s rSub { size 8{x} } ={}} {} ________
    • iii. n=________n= size 12{n} {}________
    • iv. n−1=________n−1= size 12{n - 1} {}________
  • b. Define the Random Variables XX size 12{X} {} and X¯X¯ size 12{ {overline {X}} } {}, in words.
  • c. Which distribution should you use for this problem? Explain your choice.
  • d. Construct a 92% confidence interval for the population average number of unoccupied seats per flight.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.

Exercise 13

According to a recent survey of 1200 people, 61% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job.

  • a. Define the Random Variables XX size 12{X} {} and P'P' size 12{P'} {}, in words.
  • b. Which distribution should you use for this problem? Explain your choice.
  • c. Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.

Solution 13

  • b. N ( 0 . 61 , ( 0 . 61 ) ( 0 . 39 ) 1200 ) N ( 0 . 61 , ( 0 . 61 ) ( 0 . 39 ) 1200 ) size 12{N \( 0 "." "61", { { size 8{ \( sqrt {0 "." "61"} \) \( "." "039" \) } } over { size 8{"1200"} } } \) } {}
  • c.
    • i. CI: (0.59, 0.63)
    • iii. EB = 0.02

Exercise 14

A survey of the average amount of cents off that coupons give was done by randomly surveying one coupon per page from the coupon sections of a recent San Jose Mercury News. The following data were collected: 20¢; 75¢; 50¢; 65¢; 30¢; 55¢; 40¢; 40¢; 30¢; 55¢; $1.50; 40¢; 65¢; 40¢. Assume the underlying distribution is approximately normal.

  • a.
    • i. x¯=________x¯= size 12{ {overline {x}} ={}} {}________
    • ii. s x = ________ s x = size 12{s rSub { size 8{x} } ={}} {} ________
    • iii. n=________n= size 12{n} {}________
    • iv. n−1=________n−1= size 12{n - 1} {}________
  • b. Define the Random Variables XX size 12{X} {} and X¯X¯ size 12{ {overline {X}} } {}, in words.
  • c. Which distribution should you use for this problem? Explain your choice.
  • d. Construct a 95% confidence interval for the population average worth of coupons.
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.
  • e. If many random samples were taken of size 14, what percent of the confident intervals constructed should contain the population average worth of coupons? Explain why.

Exercise 15

An article regarding interracial dating and marriage recently appeared in the Washington Post. Of the 1709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. In this survey, 86% of blacks said that their families would welcome a white person into their families. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.

  • a. We are interested in finding the 95% confidence interval for the percent of black families that would welcome a white person into their families. Define the Random Variables XX size 12{X} {} and P'P' size 12{P'} {}, in words.
  • b. Which distribution should you use for this problem? Explain your choice.
  • c. Construct a 95% confidence interval
    • i. State the confidence interval.
    • ii. Sketch the graph.
    • iii. Calculate the error bound.

Solution 15

  • b. N ( 0 . 86 , ( 0 . 86 )