Hypothesis Test for a Population Proportion
Perform a one-sample z-test for a population proportion to test a company's claim about customer satisfaction rates.
Problem Scenario
A telecommunications company claims that at least 85% of its customers are satisfied with their service. A consumer advocacy group suspects the true satisfaction rate is lower. They randomly survey 200 customers and find that 160 report being satisfied. Test the company's claim at the 0.05 significance level.
Given Data
Requirements
- State the null and alternative hypotheses in terms of the population proportion
- Calculate the test statistic and p-value
- State the conclusion in the context of the problem
Solution
Step 1:
State the hypotheses. H_0: p >= 0.85 (the satisfaction rate is at least 85%). H_a: p < 0.85 (the satisfaction rate is less than 85%). This is a left-tailed test.
Step 2:
Check conditions: np_0 = 200(0.85) = 170 >= 10 and n(1 - p_0) = 200(0.15) = 30 >= 10. Conditions are met for the z-test.
Step 3:
Calculate the standard error under H_0: SE = sqrt[p_0(1 - p_0) / n] = sqrt[(0.85)(0.15) / 200] = sqrt[0.1275 / 200] = sqrt(0.0006375) = 0.02525.
Step 4:
Calculate the z-test statistic: z = (p-hat - p_0) / SE = (0.80 - 0.85) / 0.02525 = -0.05 / 0.02525 = -1.98.
Step 5:
Find the p-value. For a left-tailed test: p-value = P(Z < -1.98) = 0.0239.
Step 6:
Decision: Since p-value (0.0239) < alpha (0.05), we reject H_0. There is sufficient evidence to conclude that the customer satisfaction rate is less than 85%.
Final Answer
z = -1.98, p-value = 0.0239. At the 0.05 significance level, we reject the company's claim. There is statistically significant evidence that the true customer satisfaction rate is below 85%.
Key Takeaways
- ✓When testing a claim about a proportion, use the claimed value (p_0) in the standard error formula, not the sample proportion (p-hat).
- ✓The sample proportion p-hat = 0.80 is 5 percentage points below the claimed 0.85. The z-test determines whether this difference is statistically significant or could be due to random sampling variability.
- ✓A significant result at alpha = 0.05 means there is less than a 5% chance of observing a sample proportion this low (or lower) if the true proportion really were 0.85.
Common Errors to Avoid
- ✗Using p-hat instead of p_0 in the standard error formula. Under the null hypothesis, we assume p = p_0, so the standard error must use p_0.
- ✗Setting up the wrong alternative hypothesis. Read the problem carefully: "suspects the rate is lower" indicates a left-tailed test (H_a: p < 0.85).
- ✗Confusing statistical significance with practical significance. A 5% drop in satisfaction (from 85% to 80%) may or may not be practically meaningful depending on the business context.
Practice More Problems with AI
Snap a photo of any problem and get instant explanations.
Download StatsIQFAQs
Common questions about this problem type
In hypothesis testing, we calculate the test statistic under the assumption that H_0 is true. Since H_0 specifies that p = p_0, the standard error is computed using p_0. This differs from confidence intervals, where we use p-hat because we are estimating the unknown proportion.
If np_0 < 10 or n(1 - p_0) < 10, the normal approximation to the binomial is not appropriate. In that case, use an exact binomial test, which calculates the p-value directly from the binomial distribution without the normal approximation.