A new medical company claims that 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.what is the critical value
To test the claim, we can use the hypothesis test for a proportion.
The null hypothesis, denoted as H0, is that the claimed proportion of satisfied workers at ABSA is 70%:
H0: p = 0.70
The alternative hypothesis, denoted as Ha, is that the claimed proportion of satisfied workers at ABSA is not 70%:
Ha: p ≠ 0.70
The significance level, denoted as α, is 10% or 0.10.
Therefore, the critical value is the z-value that corresponds to a cumulative probability of (1-α/2) or (1-0.10/2 = 0.95).
To find the critical value, we can use a standard normal distribution table or a statistical software. The critical value at 0.95 cumulative probability is approximately 1.96.
Thus, the critical value is 1.96.
To test the claim that 70% of the workers at ABSA are satisfied with their new medical scheme, we can use a hypothesis test.
The null hypothesis (H0) is that the true proportion of satisfied workers is equal to 70%:
H0: p = 0.70
The alternative hypothesis (Ha) is that the true proportion of satisfied workers is not equal to 70%:
Ha: p ≠ 0.70
To test this claim, we can use the z-test for proportions. The test statistic is calculated using the formula:
z = (p̂ - p0) / sqrt(p0(1-p0)/n)
Where:
- p̂ is the sample proportion (100/150 = 0.667)
- p0 is the null hypothesis proportion (0.70)
- n is the sample size (150)
Now, we need to find the critical value for the test. The critical value is found using the z-table for a two-tailed test at a 10% significance level.
Step-by-step:
1. Determine the significance level α. In this case, α = 0.10.
2. Find the corresponding z-score for the upper tail of α/2 (0.10/2 = 0.05).
- From the z-table, the z-score for 0.05 is approximately 1.645.
3. Find the corresponding z-score for the lower tail of α/2 (0.10/2 = 0.05).
- Since the test is two-tailed, the z-score for the lower tail is the negative of the z-score for the upper tail.
- Therefore, the z-score for the lower tail is approximately -1.645.
So, the critical values for this test are -1.645 and 1.645.
To test the claim that 70% of the workers at ABSA are satisfied with their new medical scheme, we will use a hypothesis test.
Let's set up the null and alternative hypotheses:
- Null hypothesis (H0): The proportion of workers at ABSA satisfied with their new medical scheme is 70%.
- Alternative hypothesis (Ha): The proportion of workers at ABSA satisfied with their new medical scheme is not equal to 70%.
To test these hypotheses, we will use a significance level of 10%, which means if the p-value (probability value) is less than 0.10, we will reject the null hypothesis in favor of the alternative hypothesis.
To find the critical value for this test, we need to determine the rejection regions. Since this is a two-tailed test (we're testing if the proportion is not equal to 70%), we will divide the significance level by 2, resulting in a rejection region of 0.05 on each tail.
To find the critical value, we will use a Z-table or a statistical calculator. The Z-table provides us with the Z-score associated with a given cumulative probability.
Using a Z-table, we find the z-value associated with a cumulative probability of 0.05. This corresponds to a z-value of approximately -1.645.
Therefore, the critical value for this two-tailed test at a 10% significant level is -1.645.