Chapter 11 2 Suppose That The Amount Time Teenagers Spend

19. The term 1 – refers to the level of confidence that a confidence interval does not contain the population

parameter.

20. In the formula

/ 2 /nx z





, the subscript

/ 2



refers to the area in the lower tail or upper tail of the

sampling distribution of the sample mean.

21. The larger the level of confidence used in constructing a confidence interval estimate of the population

mean, the narrower the confidence interval.

22. It is impossible to construct a confidence interval estimate of the population mean if the population

variance is unknown.

23. If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be

relatively efficient.

24. In developing an interval estimate for a population mean, the population standard deviation



was

assumed to be 8. The interval estimate was 50.0 ± 2.50. Had



equalled 16, the interval estimate

would have been 100 ± 5.0.

25. The sample mean is an unbiased estimator of the population mean



, and (when sampling from a

normal population) the sample median is also an unbiased estimator of



. However, the sample

median is relatively more efficient than the sample mean.

26. A 95% confidence interval estimate for a population mean



is determined to be 75 to 85. If the

confidence level is reduced to 80%, the confidence interval for



becomes narrower.

27. When constructing confidence interval for a parameter, we generally set the confidence level

1



−

close to 1 (usually between 0.90 and 0.99) because we would like to be reasonably confident that the

interval includes the actual value of the population parameter.

28. Suppose that a 90% confidence interval for



is given by

0.75x

. This notation means that we are

90% confident that



falls between

0.75x−

and

0.75x+

.

29. We cannot interpret the confidence interval estimate of



as a probability statement about



, simply

because the population mean is a fixed but unknown quantity.

30. The width of the confidence interval estimate of the population mean



is a function of only two

quantities, the population standard deviation



and the sample size n.

31. At a given sample size and level of confidence, the smaller the population standard deviation



, the

wider and thus the less precise the confidence interval estimate of



is.

32. In general, increasing the confidence level

1



−

will narrow the interval, and decreasing it widens the

interval.

33. Suppose that a 95% confidence interval for



is given by

3.25x

. This notation means that if we

repeatedly draw samples of the same size from the same population, 95% of the values of

x

will be

such that



would lie somewhere between

3.25x−

and

3.25x+

.

34. When constructing a confidence interval estimate of



, doubling the sample size n reduces the width

%of the interval by half.

35. You need four values to construct the confidence interval estimate of



. These are the sample mean,

sample size, population standard deviation and confidence level.

36. In an effort to identify the true proportion of first-year university students who are under 18 years of

age, a random sample of 500 first-year students was taken. Only 50 of them were under the age of 18.

The value 0.10 would be used as a point estimate to the true proportion of first-year students aged

under 18.

37. The mean of the sampling distribution of the sample proportion

ˆ

p

, when the sample size n = 100 and

the population proportion p = 0.92, is 92.0.

38. The standard error of the sampling distribution of the sample proportion

ˆ

p

, when the sample size n =

100 and the population proportion p = 0.30, is 0.0021.

39. Recall the rule of thumb used to indicate when the normal distribution is a good approximation to the

sampling distribution for the sample proportion

ˆ

p

. For the combination n = 50; p = 0.05, the rule is

satisfied.

40. As a general rule, the normal distribution is used to approximate the sampling distribution of the

sample proportion only if both n 

ˆ

p

and n 

ˆ

q

are at least five (

ˆ

q

= 1 –

ˆ

p

).

SHORT ANSWER

1. A survey of 100 retailers revealed that the mean after-tax profit was $80 000. Assuming that the

population standard deviation is $15 000, determine the 95% confidence interval estimate of the mean

after-tax profit for all retailers.

2. The nighttime temperature readings for 20 winter days in Sydney are normally distributed with a mean

of 5.5ºC and a population standard deviation of 1.5ºC. Determine the 90% confidence interval estimate

for the mean winter nighttime temperature.

3. A sample of 50 students was asked how much time they spend on average a week in front of a

computer. The sample mean and sample standard deviation were 15.8 and 2.7 hours, respectively.

Estimate with 95% confidence interval the mean number of hours students spend in front of a

computer a week.

4. A random sample of 10 waitresses in Darwin revealed the following hourly earnings (in dollars,

including tips):

19

18

15

16

18

17

16

18

20

14

If the hourly earnings are normally distributed with a standard deviation of $4.5, estimate with 95%

confidence the mean hourly earnings for all waitresses in Darwin.

5. Determine the sample size that is required to estimate a population mean to within 0.4 units with a

99% confidence when the population standard deviation is 1.75.

6. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a