# Chapter 10: Hypothesis Tests About u and p: Applications

1.  Consider this lower-tail hypothesis test: H0: μ ≥ 25 against Ha: μ < 25.  If a random sample of n=36 results in a value of x̅  = 24, what is the test statistic Z? Assume that the population standard deviation is known to be α = 3.

1. -1.645
2. -2 X
3. 2
4. 1.645

Solution:

> z <- (24 - 25) / (3 / sqrt(36))

> z

[1] -2

2.  For the previous question, what is the p-value?

1. 0.01109069
2. 0.04436276
3. 0.02275013 X
4. 0.03412520

Solution:

> pvalue <- pnorm((24 - 25) / (3 / sqrt(36)))

> pvalue

[1] 0.02275013

3. Given the p-value calculated for the previous question, what should be the correct conclusion if α = 0.01?

1. Reject H0: μ ≥ 25
2. Accept H0: μ ≥ 25
3. Do not reject Ha: μ < 25
4. Do not reject H0: μ ≥ 25 X

Solution:

Since p-value = 0.02275013 > a = 0.01, we do not reject H0: μ ≥ 25

4.  Consider this two-tail hypothesis test: H0: μ = -12 against Ha: μ ≠ -12.  If a random sample of n=49 provides a value of x̅  = -9, what is the test statistic Z? Assume that the population standard deviation is known to be α = 10.50.

1. 1.96
2. -2
3. 2 X
4. -1.96

Solution:

> z <- (-9 - (-12)) / (10.50 / sqrt(49))

> z

[1] 2

5. For the previous question, what is the p-value?

1. 0.04550026 X
2. 0.06279036
3. 0.03139518
4. 0.05232530

Solution:

> pvalue <- 2 * pnorm((-9 - (-12)) / (10.50 / sqrt(49)), lower.tail = FALSE)

> pvalue

[1] 0.04550026

6.  Given the p-value calculated for the previous question, what should be the correct conclusion if α = 0.05?

1. Accept H0: μ = -12
2. Reject H0: μ = -12 X
3. There is insufficient evidence to conclude one way or the other
4. Reject Ha: μ ≠ -12

Solution:

Since p-value = 0.04550026 < α = 0.05, we reject H0: μ = -12

7.  Consider this upper-tail hypothesis test: H0: p ≤ 0.70 against Ha: p > 0.70.  If a random sample of n = 800 provides a sample proportion 0.76 (i.e., 608 out of 800), what is the test statistic Z?

1. 2.185478
2. 2.384158
3. 3.973597 X
4. 3.377558

Solution:

> z <- (0.76 - 0.70) / (sqrt(0.76 * 0.24 / 800))

> z

[1] 3.973597

8. For the previous question, what is the p-value?

1. 0.00002300847
2. 0.00000849543
3. 0.00004247717
4. 0.00003539764 X

Solution:

> pvalue <- pnorm((0.76 - 0.70) / (sqrt(0.76 * 0.24 / 800)), lower.tail = FALSE)

> pvalue

[1] 0.00003539764

9.  Given the p-value calculated for the previous question, what should be the correct conclusion if α = 0.10?

1. Accept H0: p ≤ 0.70
2. Reject H0: p ≤ 0.70 X
3. There is insufficient evidence to conclude one way or another
4. Ha: p > 0.70

Solution:

Since p-value = 0.00003539764 < α = 0.10, we reject H0: p ≤ 0.70.

10.  Consider this upper-tail hypothesis test: H0: μ ≤ 220 against Ha: μ > 220.  Suppose a random sample of n = 100 will be selected and the population standard deviation is σ = 25.  What is the probability of making a Type II error β if the actual population mean is μ  = 227.  Use α = 0.05.

1. 0.1240152 X
2. 0.1364168
3. 0.1736213
4. 0.0868106

Solution:

> pnorm(((220 + qnorm(0.05, lower.tail = FALSE) * 25 / sqrt(100)) - 227) / (25 / sqrt(100)))

[1] 0.1240152