4.2 Hypothesis Testing for Proportions
We can use StatCrunch → Stat → Proportion Stats → One Sample → With Summary (or With Data) to test a claim about a population proportion p as long as the requirements are met:
- We have a simple random sample.
- Each sample observation can be classified as success or failure.
- The number of successes and the number of failures in the sample are each greater than or equal to 5.
- Driving and Texting: In a survey, 1864 out of 2246 randomly selected adults in the U.S. said that texting while driving should be illegal (based on data from Zogby International). Consider a hypothesis test that uses a 0.05 significance level to test the claim that more than 80% of adults believe that texting while driving should be illegal.
- Put a box around the statement of the claim in the problem.
More than 80% of adults believe that texting while driving should be illegal. - The original Claim:
\(p>0.8\) - \(H_0\):
\(p \leq 0.8\) We will assume that \(p = 0.8\) - \(H_A\):
\(p > 0.8\) - Type of Test:
Right-tailed test - Draw a graph of the critical/rejection region
- Significance Level \(\alpha\) (area of critical regions):
\(\alpha =.05\) - Sample proportion:
\(\hat{p}=\frac{1864}{2246}=.83\) - p-value:
0.0002 - Decision about the null:
Since 0.0002 is less than 0.05, we reject the null hypothesis.
- Concluding statement:
There is sufficient sample evidence to support the claim that more than 80% of adults believe texting while driving should be illegal.
Claim has \( =,\ \ \le,\ \ \geq \) Claim has \( \neq,\ <,\ \ > \) Claim is null hypothesis Claim is alternate hypothesis Reject the null hypothesis “There is sufficient sample evidence to reject the claim that…” “There is sufficient sample evidence to support the claim that…” Fail to reject the null hypothesis “There is not sufficient sample evidence to reject the claim that…” “There is not sufficient sample evidence to support the claim that…”
- Put a box around the statement of the claim in the problem.
- Cell phone and Cancer: In a study of 420,095 Danish cell phone users, 135 subjects developed cancer of the brain or nervous system (based on data from the Journal of the National Cancer Institute as reported in USA Today). Test the claim of a once popular belief that such cancers are affected by cell phone use. That is, test the claim that cell phone users develop cancer of the brain or nervous system at a rate that is different from the rate of 0.034% for people who do not use cell phones. Because this issue has such great importance, use a 0.01 significance level.
- Put a box around the statement of the claim in the problem.
Cell phone users develop cancer of the brain or nervous system at a rate that is different from the rate of 0.034% for people who do not use cell phones. - The original Claim:
\(p \neq 0.00034\) - \(H_0\):
\(p = 0.00034\) - \(H_A\):
\(p \neq 0.00034\) - Type of Test:
Two-tailed test - Draw a graph of the critical/rejection region
- Put a box around the statement of the claim in the problem.
- Significance Level \(\alpha\) (area of critical regions):
\(\alpha =0.01\) - Sample proportion:
\(\hat{p}=\frac{135}{420095} \approx 0.00032\) - p-value:
0.5122 - Decision about the null:
Since 0.5122 is greater than 0.01, we fail to reject the null hypothesis - Concluding statement:
There is not sufficient sample evidence to support the claim that cell phone users develop cancer of the brain/nervous system at a rate different from non-cell phone users.
Claim has \( =,\ \ \le,\ \ \geq \) | Claim has \( \neq,\ <,\ \ > \) | |
---|---|---|
Claim is null hypothesis | Claim is alternate hypothesis | |
Reject the null hypothesis | “There is sufficient sample evidence to reject the claim that…” | “There is sufficient sample evidence to support the claim that…” |
Fail to reject the null hypothesis | “There is not sufficient sample evidence to reject the claim that…” | “There is not sufficient sample evidence to support the claim that…” |
- Put a box around the statement of the claim in the problem.
No more than 2.2% of this drug’s users experience flu-like symptoms as a side effect. - Claim:
\(p \leq 0.022\) - \(H_0\):
\(p \leq 0.022\) - \(H_A\):
\(p > 0.022\) - Type of Test:
Right-tailed test - Draw a graph of the critical/rejection region
- Significance Level \(\alpha\) (area of critical regions):
\(\alpha =0.1\) - p-value:
0.0061 - Decision about the null:
Since 0.0061 is less than 0.10, we reject the null hypothesis. - Concluding statement:
There is sufficient sample evidence to reject the claim that no more than 2.2% of the drug’s users experience flu-like symptoms.
Claim has \( =,\ \ \le,\ \ \geq \) | Claim has \( \neq,\ <,\ \ > \) | |
---|---|---|
Claim is null hypothesis | Claim is alternate hypothesis | |
Reject the null hypothesis | “There is sufficient sample evidence to reject the claim that…” | “There is sufficient sample evidence to support the claim that…” |
Fail to reject the null hypothesis | “There is not sufficient sample evidence to reject the claim that…” | “There is not sufficient sample evidence to support the claim that…” |
Goldfish Smile Hypothesis Testing
One possible choice is to claim that p > .50. The solutions on the rest of the activity will be based on this claim. However, this is not what you are required to choose when you complete this in class.
H0: _______________
Ha: _______________
\( \alpha = .05\)
- Number of goldfish in your sample: n = _______________
100 - Number of SMILING goldfish in your sample: x = _______________
35 - \( \hat{p}=\frac{x}{n} \) (the sample proportion) = _______________
.35
There is not sufficient evidence to support the claim that the proportion of all goldfish with smiles is greater than 50%.
Confidence Intervals
- Number of goldfish in your sample: n = _______________
100 - Number of SMILING goldfish in your sample: x = _______________
35 - \( \hat{p}=\frac{x}{n} \) (the sample proportion) = _______________
.35
- Best Point Estimate: _________________________
0.35
\( (0.257, 0.443) \)
E = ______________________
In a hypothesis test for a proportion, we can also calculate a critical value and a test statistic.
- The critical value is the z-score associated with the significance level α. It defines the critical region.
- The test statistic is the z-score associated with the p-value. The test statistic is a standardized value calculated from the sample. If the test statistic is in the critical region, then we reject the null hypothesis.
- Put a box around the statement of the claim in the problem.
More than 15% of sneakers that are sold attach with Velcro. - Claim:
\(p > 0.15\) - \(H_0\):
\(p \leq 0.15\) - \(H_A\):
\(p > 0.15\) - Type of Test:
Right-tailed test - Draw a graph of the critical/rejection region
- Significance Level \(\alpha\) (area of critical regions):
\(\alpha =0.10\) - Sample proportion:
10/35 = 0.286 - Critical value (label this on your graph)
1.64 - j. Test statistic (label this on your graph)
2.25 If the test statistic is in the critical region, then reject the null hypothesis.
- Decision about the null:
Reject the null hypothesis. - Concluding statement:
There is sufficient evidence to support the claim that more than 15% of sneakers that are sold attach with Velcro.
Claim has \( =,\ \ \le,\ \ \geq \) | Claim has \( \neq,\ <,\ \ > \) | |
---|---|---|
Claim is null hypothesis | Claim is alternate hypothesis | |
Reject the null hypothesis | “There is sufficient sample evidence to reject the claim that…” | “There is sufficient sample evidence to support the claim that…” |
Fail to reject the null hypothesis | “There is not sufficient sample evidence to reject the claim that…” | “There is not sufficient sample evidence to support the claim that…” |