COMPREHENSIVE TUTORIAL_III



COMPREHENSIVE TUTORIAL_III

Estimation: Point and Interval Estimation, Estimator and Estimates, Confidence Intervals, Interval Estimates of Mean and Proportion from Large Samples, Interval Estimation Using t Distribution, Sample Size for Estimating Means and Proportions

1. Define a point estimate : Answer : A point estimate is a single valued estimate of a population parameter.

2. Define an Esimator & an Estimate: Estimator is a sample statistics used to estimate population parameter and Estimate is a specific observed value of an estimator

3. Define an interval estimate : Answer : An interval estimate is an interval within which the population parameter is likely to lie.

4. Define level of confidence : Answer : It is the probability ( generally .90,.95,.99) that the interval estimate will cover the true value of the population of interest

5. Define confidence interval: Answer: It is an interval estimate obtained by a procedure satisfying the probability requirement. It is found by constructing an interval around the point estimate of the form :

Point Estimate +/- Multiple ( Estimated standard deviation of point estimate)

where the multiple is often a normal distribution percentage point or a

t-distribution percentage point.

6. Give point estimates of population mean (µ ) , population standard deviation (( ) and population proportion ( p )

7. Give formulae for sample size for estimating means & proportions.

8. when t- distribution is used for interval estimation. It is used when the population is normally distributed, population s.d is unknown & sample size is less than 30.

9. What are the four desirable properties of point estimates? Describe each.

10. In point estimation

a. data from the population is used to estimate the population parameter

b. data from the sample is used to estimate the population parameter

c. data from the sample is used to estimate the sample statistic

d. the mean of the population equals the mean of the sample

Answer: b

11. The sample statistic s is the point estimator of

a. (

b. (

c. [pic]

d. [pic]

Answer: b

12. The sample mean is the point estimator of

a. (

b. (

c. [pic]

d. [pic]

Answer: a

13. A sample statistic is an unbiased estimator of the population parameter if

a. the expected value of the sample statistic is equal to zero

b. the expected value of the sample statistic is equal to one

c. the expected value of the sample statistic is equal to the population parameter

d. it is equal to zero

Answer: c

14. A property of a point estimator that occurs whenever larger sample sizes tend to provide point estimates closer to the population parameter is known as

a. efficiency

b. unbiased sampling

c. consistency

d. relative estimation

Answer: c

15. The standard deviation of a point estimator is called the

a. standard deviation

b. standard error

c. point estimator

d. variance of estimation

Answer: b

16. The sample statistic, such as [pic], s, or [pic], that provides the point estimate of the population parameter is known as

a. a point estimator

b. a parameter

c. a population parameter

d. a population statistic

Answer: a

17. A simple random sample of 5 observations from a population containing 400 elements was taken, and the following values were obtained.

12 18 19 20 21

A point estimate of the mean is

a. 400

b. 18

c. 20

d. 10

Answer: b

18. Given two unbiased point estimators of the same population parameter, the point estimator with the smaller variance is said to have

a. smaller relative efficiency

b. greater relative efficiency

c. smaller consistency

d. larger consistency

Answer: b

19. Whenever the estimation process summarizes all of the information a sample has about a population parameter, the point estimator has the property of

a. relative consistency

b. full consistency

c. sufficiency

d. insufficiency

Answer: c

20. The following data was collected from a simple random sample of a population.

13 15 14 16 12

The point estimate of the population standard deviation is

a. 2.500

b. 1.581

c. 2.000

d. 1.414

Answer: b

21. Four hundred people were asked whether gun laws should be more stringent. Three hundred said "yes," and 100 said "no." The point estimate of the proportion in the population who will respond "yes" is

a. 300

b. approximately 300

c. 0.75

d. 0.25

Answer: c

22. Starting salaries of a sample of five management majors along with their genders are shown below.

Salary

Employee (in $1,000s) Gender

1 30 F

2 28 M

3 22 F

4 26 F

5 19 M

a. What is the point estimate for the starting salaries of all management majors?

b. Determine the point estimate for the variance of the population.

c. Determine the point estimate for the proportion of male employees.

Answers:

a. 25 (thousands)

b. 20 (thousands)

c. 0.4

23. In order to determine an interval for the mean of a population with unknown standard deviation a sample of 25 items is selected. The mean of the sample is determined to be 23. The number of degrees of freedom for reading the t value is

a. 22

b. 23

c. 24

d. 25

Answer: c

24. If we want to provide a 95% confidence interval for the mean of a population, the confidence coefficient is

a. 0.485

b. 1.96

c. 0.95

d. 1.645

Answer: c

25. As the number of degrees of freedom for a t distribution increases, the difference between the t distribution and the standard normal distribution

a. becomes larger

b. becomes smaller

c. stays the same

d. None of these alternatives is correct.

Answer: b

26. For the interval estimation of ( when ( is known and the sample is large, the proper distribution to use is

a. the normal distribution

b. the t distribution with n degrees of freedom

c. the t distribution with n - 1 degrees of freedom

d. the t distribution with n - 2 degrees of freedom

Answer: a

27. An estimate of a population parameter that provides an interval of values believed to contain the value of the parameter is known as the

a. confidence level

b. interval estimate

c. parameter value

d. population estimate

Answer: b

28. In developing an interval estimate, if the population standard deviation is unknown

a. it is impossible to develop an interval estimate

b. the standard deviation is arrived at using historical data

c. the sample standard deviation can be used

d. it is assumed that the population standard deviation is 1

Answer: c

29. When the level of confidence increases, the confidence interval

a. stays the same

b. becomes wider

c. becomes narrower

d. becomes narrower for small sample sizes

Answer: b

30. A random sample of 144 observations has a mean of 20, a median of 21, and a mode of 22. The population standard deviation is known to equal 4.8. The 95.44% confidence interval for the population mean is

a. 15.2 to 24.8

b. 19.200 to 20.800

c. 19.216 to 20.784

d. 21.2 to 22.8

Answer: b

31. A random sample of 16 students showed an average age of 25 years and a standard deviation of 2 years. The 98% confidence interval for the true average age of students is

a. 24.329 to 26.67

b. 23.699 to 26.301

c. 24.487 to 25.513

d. 24.316 to 25.684

Answer: b

32. The sample size needed to provide a margin of error of 2 or less with a .95 probability when the population standard deviation equals 11 is

a. 10

b. 11

c. 116

d. 117

Answer: d

33. Which of the following best describes the form of the sampling distribution of the sample proportion?

a. When standardized, it is exactly the standard normal distribution.

b. When standardized, it is the t distribution.

c. It is approximately normal as long as n ( 30.

d. It is approximately normal as long as np ( 5 and n(1-p) ( 5.

Answer: d

34. In a random sample of 144 observations, [pic] = 0.9. The 95% confidence interval for P is

a. 0.851 to 0.949

b. 0.876 to 0.924

c. 0.898 to 0.902

d. 0.1 to 0.9

Answer: a

35. A random sample of 100 people was taken. Eighty of the people in the sample favored Candidate A. The 95% confidence interval for the true proportion of people who favors Candidate A is

a. 0.722 to 0.878

b. 0.762 to 0.838

c. 78.04 to 81.96

d. 62.469 to 97.531

Answer: a

36. A machine that produces a major part for an airplane engine is monitored closely. In the past, 10% of the parts produced would be defective. With a .95 probability, the sample size that needs to be taken if the desired margin of error is .05 or less is

a. 7

b. 33

c. 138

d. 139

Answer: d

37. A sample of 100 cans of coffee showed an average weight of 13 ounces with a standard deviation of 0.8 ounces.

a. Construct a 95% confidence interval for the mean of the population.

b. Construct a 95.44% confidence interval for the mean of the population.

c. Discuss why the answers in Parts a and b are different.

Answers:

a. 12.8432 to 13.1568

b. 12.84 to 13.16

c. As the level of confidence increases, the confidence interval becomes wider.

38. A random sample of 121 checking accounts at a bank showed an average daily balance of $280. The standard deviation is known to be $60.

a. Is it necessary to know anything about the shape of the distribution of the account balances in order to make an interval estimate of the mean of all the account balances? Explain.

b. Find the standard error of the mean.

c. Give a point estimate of the population mean.

d. Construct 80% and 90% confidence interval estimates for the mean.

Answers:

a.No, since the sample means will be normally distributed by the central limit theorem.

b. 5.4545

c. 280

d. 273.02 to 286.98 271.05 to 288.95

39. The makers of a soft drink want to identify the average age of its consumers. A sample of 16 consumers is taken. The average age in the sample was 22.5 years with a standard deviation of 5 years.

a. Construct a 95% confidence interval for the true average age of the consumers.

b. Construct an 80% confidence interval for the true average age of the consumers.

c. Discuss why the 95% and 80% confidence intervals are different.

Answers:

a. 19.836 to 25.164

b. 20.824 to 24.176

c. As the level of confidence increases, the confidence interval gets wider.

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