Given:
- Claim: less than 70% of workers at ABSA are satisfied with their new medical scheme
- Sample size: n = 150
- Number of workers satisfied with the medical scheme: x = 100
- Significance level: α = 0.10 (10%)
To test the claim, we will use the one-sample proportion test.
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): p ≥ 0.70 (70%)
- Alternative hypothesis (Ha): p < 0.70 (70%)
Step 2: Select a significance level (α):
- Given that the significant level is 10%, α = 0.10
Step 3: Calculate the test statistic:
- The test statistic for the one-sample proportion test is the z-score.
- The formula to calculate the z-score is:
z = (p̂ - p) / √[(p * (1-p)) / n]
where p̂ is the sample proportion and p is the hypothesized population proportion.
In this case:
- p̂ = x / n = 100 / 150 = 0.67 (67%)
- p = 0.70 (70%)
- n = 150
Plugging the values into the formula, we can calculate the test statistic z:
z = (0.67 - 0.70) / √[(0.70 * (1-0.70)) / 150]
z = -0.03 / √[(0.70 * 0.30) / 150]
z = -0.03 / √(0.21 / 150)
z = -0.03 / √(0.0014)
z ≈ -0.03 / 0.0374
z ≈ -0.80
Step 4: Determine the critical value(s):
- Since the alternative hypothesis is p < 0.70, we have a one-tailed test.
- The critical value for a one-tailed test at a 10% significance level is z = -1.28.
Step 5: Make the decision:
- If the test statistic (z) is less than the critical value (-1.28), reject the null hypothesis.
- If the test statistic (z) is greater than or equal to the critical value (-1.28), fail to reject the null hypothesis.
In this case, the test statistic z is -0.80, which is greater than -1.28. Therefore, we fail to reject the null hypothesis.
Step 6: State the conclusion:
- Based on the sample data and the test, there is not enough evidence to support the claim that less than 70% of workers at ABSA are satisfied with their new medical scheme at a 10% significance level.
A new medical company claims that less than 70% of the workers at ABSA are satisfied with their new medical scheme. A random sample of 150 workers showed that 100 are satisfied with their new medical scheme. Test the claim at a 10% significant level.
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