4. In testing the hypotheses
50:
0=
H
.
50:
1
H
.
we found that the standardised test statistic is z = –1.59. Calculate the p-value.
5. Suppose that 10 observations are drawn from a normal population whose variance is 64. The
observations are:
13
21
15
19
35
24
14
18
27
30
Test at the 10% level of significance to determine whether there is enough evidence to conclude that
the population mean is greater than 20.
6. In testing the hypotheses
20:
0=
H
.
20:
1
H
.
the following information was given:
8.1, 100, 18.1, 0.025n x
= = = =
.
a. Calculate the value of the test statistic.
b. Set up the rejection region.
c. Determine the p-value.
d. Interpret the result.
26:
0=
H
26:
1
H
7. In testing the hypotheses
60:
0=
H
.
60:
1
H
.
the following information was given:
5, 100, 59, 0.05n x
= = = =
.
a. Calculate the value of the test statistic.
b. Set up the rejection region.
c. Determine the p-value.
d. Interpret the result.
8. Formulate the null and alternative hypotheses for each of the following statements:
a. The average Australian drinks 2.5 cups of coffee per day.
b. A researcher at the University of Newcastle is looking for evidence to conclude that the average
entrance score for entering first years is well over 1650.
c. The manager of the University of Tasmania bookstore claims that the average student spends less
than $400 per semester at the university’s bookstore.
9. Determine the p-value associated with each of the following values of the standardised test statistic z.
a. Two-tail test, z = 1.50.
b. One-tail test, z = 1.05.
c. One-tail test, z = –2.40.
10. For each of the following pairs of null and alternative hypotheses, determine whether or not the pair
would be appropriate for testing a hypothesis.
a.
25:
0=
H
.
25:
1
H
.
b.
30:
0=
H
.
30:
1
H
.
c.
:
0
H
35=x
.
35:
1
xH
.
d.
40:
0=xH
.
40:
1xH
.
e.
50:
0=
H
.
50:
1
H
.
11. For each of the following statements, formulate appropriate null and alternative hypotheses and
indicate whether the appropriate test will be one- or two-tailed.
a. The average tertiary entrance score for international students is well above 90.
b. The average Australian adult drinks fewer than 3 cups of coffee per day.
c. The average housewife works more than 40 hours per week in house-related activities.
d. The average employee calls in sick 3 times a year.
12. Suppose that nine observations are drawn from a normal population whose standard deviation is 2. The
observations are:
15
9
13
11
8
12
11
7
10
At 95% confidence, you want to determine whether the mean of the population from which this sample
was taken is significantly different from 10.
a. State the null and alternative hypotheses.
b. Compute the value of the test statistic.
c. Compute the p-value.
d. Interpret the results.
13. In testing the hypotheses:
40:
0=
H
,40:
1
H
the following information was given:
5.5, 25, 42, 0.10n x
= = = =
, and the sampled
population is normally distributed.
a. Calculate the value of the test statistic.
b. Set up the rejection region.
c. Determine the p-value.
d. Interpret the result.
14. In our justice system, judges instruct juries to find the defendant guilty only if there is evidence
‘beyond a reasonable doubt’. In general, what would be the result if judges instructed juries:
a. to compromise between Type I and Type II errors?
b. never to commit a Type I error?
c. never to commit a Type II error?
15.
16. In testing the hypotheses:
H0 : μ = 22.
H1 : μ < 22
the following information was given:
= 15, n = 50, x-bar = 17.5,
= 0.04.
a. Calculate the value of the test statistic.
b. Set up the rejection region.
c. Determine the p-value.
d. Interpret the result.
17. In testing the hypotheses
4.24:
0=
H
.
4.24:
1
H
.
the following information was given:
7.6, 60, 25.52, 0.06n x
= = = =
.
a. Calculate the value of the test statistic.
b. Set up the rejection region.
c. Determine the p-value.
d. Interpret the result.
18. The admissions officer for the graduate programs at the University of Melbourne believes that the
average score on an exam at his university is significantly higher than the national average of 1300.
Assume that the population standard deviation is 125 and that a random sample of 25 scores had an
average of 1375.
a. State the appropriate null and alternative hypotheses.
b. Calculate the value of the test statistic and set up the rejection region. What is your conclusion?
c. Calculate the p-value.
d. Does the p-value confirm the conclusion in part (b)?
19. With the following p–values, would you reject or fail to reject the null hypothesis? What would you
say about the test?
a. p-value = 0.0025.
b. p-value = 0.0328.
c. p-value = 0.0795.
d. p-value = 0.1940.
20. To test the hypotheses
40:
0=
H
40:
1
H
we draw a random sample of size 16 from a normal population whose standard deviation is 5. If we set
0.01,
=
find
when
37=
.
21. Calculate the probability of a Type II error for the following test of hypothesis:
50:
0=
H
50:
1
H
given that
55=
,
0.05, 10,
= =
and n = 16.
22. In testing the hypotheses:
25:
0=
H
25:
1
H
,
a random sample of 36 observations drawn from a normal population whose standard deviation is 10
produced a mean of 22.8.
Compute the value of the test statistic and specify the rejection region associated with the 5%
significance level.
23. In testing the hypotheses:
25:
0=
H
25:
1
H
,
a random sample of 36 observations drawn from a normal population whose standard deviation is 10
produced a mean of 22.8.
Compute the p-value.
24. In testing the hypotheses:
25:
0=
H
25:
1
H
,
a random sample of 36 observations drawn from a normal population whose standard deviation is 10
produced a mean of 22.8.
Can we conclude at the 5% significance level that the population mean is 25?
25. In testing the hypotheses:
25:
0=
H
25:
1
H
,
a random sample of 36 observations drawn from a normal population whose standard deviation is 10
produced a mean of 22.8.
Develop a 95% confidence interval estimate of the population mean.
26. In testing the hypotheses:
25:
0=
H
25:
1
H
,
a random sample of 36 observations drawn from a normal population whose standard deviation is 10
produced a mean of 22.8.
Explain briefly how to use the confidence interval in the previous question to test the hypothesis.
,1000=
0.10,
=
200=
27. Assume that
,1000=
0.10,
=
200=
, and n = 25. Calculate
, the probability of a Type II
error.
28. Calculate the power of the test.
29. Interpret the meaning of the power in the previous question.
30. Recalculate
if n is increased from 25 to 40.
31. What is the effect of increasing the sample size on the value of
?
32. Recalculate
if
is lowered from 0.10 to 0.05.
33. What is the effect of decreasing the significance level on the value of
?
34. During the Gulf War, a government official claimed that the average car owner refilled the fuel tank
when there was more than 3 litres of petrol left. To check the claim, 10 cars were surveyed as they
entered a service station. The amount of petrol (in litres) was measured and recorded as shown below.
3
5
3
2
3
3
2
6
4
1
Assume that the amount of petrol remaining in the tanks is normally distributed with a standard
deviation of 1 litre.
Can we conclude at the 10% significance level that the official was correct?
35. During the Gulf War, a government official claimed that the average car owner refilled the fuel tank
when there was more than 3 litres of petrol left. To check the claim, 10 cars were surveyed as they
entered a service station. The amount of petrol (in litres) was measured and recorded as shown below.
3
5
3
2
3
3
2
6
4
1
Assume that the amount of petrol remaining in the tanks is normally distributed with a standard
deviation of 1 litre.
a. Calculate the p-value.
b. Compute the probability of a Type II error if the true average amount of gas remaining in tanks is
3.5 litres.
36. A random sample of 100 families in a large city revealed that on the average these families have been
living in their current homes for 35 months. From previous analyses, we know that the population
standard deviation is 30 months.
Can we conclude at the 5% significance level that the true mean number of months families in this city
have been living in their current homes is at least 30 months?
37. A random sample of 100 families in a large city revealed that on the average these families have been
living in their current homes for 35 months. From previous analyses, we know that the population
standard deviation is 30 months.
Calculate the p-value.
38. A random sample of 100 families in a large city revealed that on the average these families have been
living in their current homes for 35 months. From previous analyses, we know that the population
standard deviation is 30 months.
Compute the probability of a Type II error if the true mean number of months families in this city have
been living in their current homes is 29.
39. A random sample of 100 families in a large city revealed that on the average these families have been
living in their current homes for 35 months. From previous analyses, we know that the population
standard deviation is 30 months.
Determine β.
40. A random sample of 100 families in a large city revealed that on the average these families have been
living in their current homes for 35 months. From previous analyses, we know that the population
standard deviation is 30 months.
a. Recalculate β if n = 100.
b. What is the effect of increasing the sample size on the value of β?
41. A random sample of 100 families in a large city revealed that on the average these families have been
living in their current homes for 35 months. From previous analyses, we know that the population
standard deviation is 30 months.
a. Recalculate if is lowered from 0.05 to 0.01.
b. What is the effect of decreasing the significance level on the value of ?
42. A random sample of 100 families in a large city revealed that on the average these families have been
living in their current homes for 35 months. From previous analyses, we know that the population
standard deviation is 30 months.
a. Calculate the power of the test.
b. Interpret your answer to part (a).
43. During the past few weeks, Bree visited Hungry Jacks fast food restaurant five times, and each time
she ordered a large-sized order of French fries. Having nothing better to do, she counted how many
French fries she received each time. The results were:
73
75
83
68
78
Assume that the number of French fries served at Hungry Jacks is normally distributed. Can we infer
at the 10% significance level that the average number of fries served in a large-sized order of French
fries at Hungry Jacks is over 70?