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.
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.
To test the claim that less than 70% of the workers at ABSA are satisfied with their new medical scheme, we can use a hypothesis test.
Step 1: State the null and alternative hypotheses.
Null hypothesis (H0): p = 0.70 (70% of workers at ABSA are satisfied with their new medical scheme)
Alternative hypothesis (H1): p < 0.70 (less than 70% of workers at ABSA are satisfied with their new medical scheme)
Step 2: Set the significance level (α).
The significance level (α) is given as 10%, which means there is a 10% chance of rejecting the null hypothesis when it is actually true. For this test, α = 0.10.
Step 3: Calculate the test statistic.
To test the claim, we need to calculate the test statistic, which in this case is the z-score. The formula for calculating the z-score for a proportion is:
z = (p - P) / sqrt((P(1 - P))/n)
where p is the sample proportion, P is the population proportion, and n is the sample size.
In this case, p = 100/150 = 0.6667 (66.67%), P = 0.70, and n = 150. Plugging these values into the formula, we get:
z = (0.6667 - 0.70) / sqrt((0.70(1 - 0.70))/150)
Calculating this gives us the test statistic z.
Step 4: Calculate the critical value.
Since the alternative hypothesis is one-tailed (less than), we need to find the critical value corresponding to the given significance level and test type. To find this critical value, we can use a standard normal distribution table or a calculator. For a 10% significance level, the critical value is -1.28 (approximately).
Step 5: Compare the test statistic with the critical value.
If the test statistic falls in the rejection region (i.e., less than the critical value), we can reject the null hypothesis.
If the test statistic is greater than the critical value, we fail to reject the null hypothesis.
Step 6: Make a decision.
If the test statistic is less than -1.28, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
I hope this helps! Let me know if you have any other questions.