Tutorial 6 (Estimation)
The required sample size can be found to reach a desired margin of error (e) with a
specified level of confidence (1 - )
σ
Given sampling error (margin of error), e  Z / 2
.
n
1. A survey of 30 adults found that the mean age of a person’s primary vehicle is 5.6 years.
Assuming the standard deviation of the population is 0.8 year; find the 90% confidence
interval of the population mean. ~ (5.36, 5.84) years.
2. The college president asks the statistics teacher to estimate the average age of the students at
their college. The statistics teacher would like to be 99% confident that the estimate should be
accurate within 1 year. From a previous study, the standard deviation of the ages is known to
be 3 years. How large a sample is necessary? ~ (60) students
3. A random sample of 6 items taken from a normal population with mean μ and variance 4.5
cm2 gave the following data:
Samples data: 12.9 cm, 13.2 cm, 14.6 cm, 12.6 cm, 11.3 cm, 10.1 cm.
a. Find the 95% confidence interval for μ ~ (10.75, 14.15) cm
b. What is the width of this confidence interval? ~ (3.4) cm
4. In a random sample of 100 batteries produced by a certain method, the average lifetime was
150 hours and the standard deviation was 25 hours
a. Find a 95% confidence interval for the mean life time of batteries produced by this method.
~ (145.1, 154.9) hours
b. Find a 99% confidence interval for the mean life time of batteries produced by this method
~ (143.56, 156.44) hours
c. An engineer claims that the mean life time is between 147 and 153 hours. With what level
of confidence can this statement be made? ~ (77%)
d. Approximately how many batteries must be sampled so that a 95% confidence interval will
specify the mean to within  2 hours? ~ (601) batteries
e. Approximately how many batteries must be sampled so that a 99% confidence interval will
specify the mean to within ±2 hours? ~ (1037) batteries
5. A chemist made 8 independent measurements of the melting point of tungsten. She obtained a
sample mean of 3410.14oC and a sample standard deviation of 1.018oC.
a. Find a 95% confidence interval for the melting point of tungsten ~ (3409.29, 3410.99) oC
b. Find a 99% confidence interval for the melting point of tungsten ~ (3408.88, 3411.40) oC
c. If the eight measurements had been 3409.76, 3409.80, 3412.66, 3409.79, 3409.76,
3409.77, 3409.80, and 3409.78, would the confidence intervals found in parts (a) and (b)
be valid? Explain. ~ (valid: the mean of the sample is between the interval)
6. A university dean whishes to estimate the average number of hours his part-time instructors
teach per week. The standard deviation from a previous study is 2.6 hours. How large a
sample must be selected if he wants to be 99% confident for finding whether the true mean
differs from the sample mean by 1 hour? ~ (45) instructors
7. A recent study of 28 city residents showed that the mean of time they had lived at their
present address was 9.3 years. The standard deviation of the sample was 2 years. Find the
90% confidence interval of the true mean. ~(8.66, 9.94) years
8. If you were constructing a 99% confidence interval of the population mean based on a sample
of n=25 where the standard deviation of the sample s = 0.05, the critical value of t will be
~ (2.797)
9. A major department store chain is interested in estimating the average amount its credit card
customers spent on their first visit to the chain’s new store in the mall. Fifteen credit card
accounts were randomly sampled and analyzed with the following results: X  $50.50 and
2
s  400 . Construct a 95% confidence interval for the average amount its credit card
customers spent on their first visit to the chain’s new store in the mall assuming that the
amount spent follows a normal distribution. ~ $(39.45, 61.55)
10. As an aid to the establishment of personnel requirements, the director of a hospital wishes to
estimate the mean number of people who are admitted to the emergency room during a 24hour period. The director randomly selects 64 different 24-hour periods and determines the
number of admissions for each. For this sample, X  19.8 and s2 = 25. Using the sample
standard deviation as an estimate for the population standard deviation, what size sample
should the director choose if she wishes to estimate the mean number of admissions per 24hour period to within 1 admission with 99% reliability,? ~ (n = 166)
11. The managers of a company are worried about the morale of their employees. In order to
determine if a problem in this area exists, they decide to evaluate the attitudes of their
employees with a standardized test. They select the Fortunato test of job satisfaction, which
has a known standard deviation of 24 points.
a. Referring to the satement, they should sample ________ employees if they want to
estimate the mean score of the employees within 5 points with 90% confidence. ~ (63)
b. Referring to the statement, due to financial limitations, the managers decide to take a
sample of 45 employees. This yields a mean score of 88.0 points. A 90% confidence
interval would go from ________ to ________. ~ (82.12 , 93.88) points.
The (1-  )100% Confidence Interval for p for Large Samples (n  30)
pˆ  z
2
pˆ 1  pˆ 
n
12. The president of a university would like to estimate the proportion of the student
population that owns a personal computer. In a sample of 500 students, 417 own a personal
computer.
a. Referring to Table 8-10, the critical value for a 99% confidence interval for this sample
is _______. ~ (2.5758)
b. Referring to Table 8-10, a 99% confidence interval for the proportion of student
population who own a personal computer is from _____ to ______. ~ (0.7911, 0.8769)
c. Referring to Table 8-10, the sampling error of a 99% confidence interval for the
proportion of student population who own a personal computer is _______. ~ (0.04286)
13.
A confidence interval was used to estimate the proportion of statistics students that are
female. A random sample of 72 statistics students generated the following 90% confidence
interval: (0.438, 0.642). Using the information above, what total size sample would be
necessary if we wanted to estimate the true proportion to within 0.08 using 95%
confidence? ~ (150)
14.
On a certain day, a large number of fuses were manufactured, each rated at 15A. A
sample of 75 fuses is drawn from the day’s production, and 17 of them were found to have
burnout amperages greater than 15A.
a. Find a 95% confidence interval for the proportion of fuses manufactured that day
whose burnout amperage is greater than 15A. ~ (0.1319, 0.3215)
b. Find a 98% confidence interval for the proportion of fuses manufactured that day
whose burnout amperage is greater than 15A. ~ (0.1143, 0.3391)
c. Find the sample size needed for a 95% confidence interval to specify the proportion to
within ±0.05. ~ (270) fuses
d. Find the sample size needed for a 98% confidence interval to specify the proportion to
within ±0.05 ~ (380) fuses
15. A random sample of 400 electronic components manufactured by a certain process are
tested, and 30 are found to be defective.
a. Let p represent the proportion of components manufactured by this process that are
defective. Find a 95% confidence interval for p. ~(0.0492, 0.1008)
b. How many components must be sampled so that the 95% confidence interval will
specify the proportion defective to within ±0.02? ~(667) components
16. A recent study of 180 females showed that 25% took vitamins. A medical researcher
whishes to determine the percentage of females who take vitamins. He wishes to be 99%
confident that the estimate is within 2 percentage points of the true proportion. How large
should the sample size be? ~ (3111) females
17. A previous study found that 18% of the 100 people surveyed said that they did snack
before bedtime. A nutritionist wishes to determine, within 2%, the true proportion of
adults who snack before bedtime. If she wishes to be 95% confident that her estimate
contains the population proportion, how large a sample will she need? ~ (1418) peoples
18. The tobacco industry closely monitors all surveys that involve smoking. One survey
showed that among 785 randomly selected students who completed four years of college,
18.3% are smokers. Construct the 98% confidence interval for the true percentage of
smokers among all students who completed four years of college. ~(0.151, 0.215)
19. A university dean is interested in determining the proportion of students who receive
some sort of financial aid. Rather than examine the records for all students, the dean
randomly selects 200 students and finds that 118 of them are receiving financial aid. Use
a 90% confidence interval to estimate the true proportion of students who receive
financial aid. If the dean wanted to estimate the proportion of all students receiving
financial aid to within 3% with 99% reliability, how many students would need to be
sampled? ~ (0.53, 0.65) , (n = 1,784) students
20. The head of a computer science department is interested in estimating the proportion of
students entering the department who will choose the new computer engineering option.
Suppose there is no information about the proportion of students who might choose the
option. What size sample should the department head take if she wants to be 95%
confident that the estimate is within 0.10 of the true proportion? ~ (97)
21. The head of a computer science department is interested in estimating the proportion of
students entering the department who will choose the new computer engineering option.
A preliminary sample indicates that the proportion will be around 0.25. Therefore, what
size sample should the department head take if she wants to be 95% confident that the
estimate is within 0.10 of the true proportion? ~ (73)