Statistics with R
Student Resources
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.
- -1.645
- -2 X
- 2
- 1.645
Solution:
> z <- (24 - 25) / (3 / sqrt(36))
> z
[1] -2
2. For the previous question, what is the p-value?
- 0.01109069
- 0.04436276
- 0.02275013 X
- 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?
- Reject H0: μ ≥ 25
- Accept H0: μ ≥ 25
- Do not reject Ha: μ < 25
- 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.
- 1.96
- -2
- 2 X
- -1.96
Solution:
> z <- (-9 - (-12)) / (10.50 / sqrt(49))
> z
[1] 2
5. For the previous question, what is the p-value?
- 0.04550026 X
- 0.06279036
- 0.03139518
- 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?
- Accept H0: μ = -12
- Reject H0: μ = -12 X
- There is insufficient evidence to conclude one way or the other
- 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?
- 2.185478
- 2.384158
- 3.973597 X
- 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?
- 0.00002300847
- 0.00000849543
- 0.00004247717
- 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?
- Accept H0: p ≤ 0.70
- Reject H0: p ≤ 0.70 X
- There is insufficient evidence to conclude one way or another
- 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.
- 0.1240152 X
- 0.1364168
- 0.1736213
- 0.0868106
Solution:
> pnorm(((220 + qnorm(0.05, lower.tail = FALSE) * 25 / sqrt(100)) - 227) / (25 / sqrt(100)))
[1] 0.1240152