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.
To test the claim, we need to set up the null and alternative hypotheses.
Let p be the true proportion of workers at ABSA who are satisfied with their new medical scheme.
Null hypothesis: p = 0.70
Alternative hypothesis: p ≠ 0.70 (two-tailed test)
Next, we can calculate the test statistic. The test statistic for testing proportions is the z-score, which can be computed as:
z = (p̂ - p) / √[(p * (1-p)) / n]
where p̂ is the sample proportion, p is the hypothesized proportion, and n is the sample size.
In this case, p̂ = 100/150 = 0.6667, p = 0.70, and n = 150.
Plugging in these values, we get:
z = (0.6667 - 0.70) / √[(0.70 * (1-0.70)) / 150]
= -0.0333 / √[(0.70 * 0.30) / 150]
= -0.0333 / 0.0322
= -1.0311
Now, we need to compare this z-score to the critical value(s) to determine if we can reject the null hypothesis.
At a 10% level of significance, we perform a two-tailed test, so we divide the significance level by 2 to get 0.05 for each tail.
The critical z-score for a 0.05 significance level is approximately ±1.96.
Since -1.0311 is not less than -1.96 or greater than 1.96, we do not have enough evidence to reject the null hypothesis.
Therefore, we fail to reject the claim that 70% of the workers at ABSA are satisfied with their new medical scheme at a 10% significant level.
To test the claim, we can use the hypothesis testing approach. Here are the steps:
Step 1: State the null hypothesis (H0) and the alternative hypothesis (H1):
- Null hypothesis (H0): The proportion of workers satisfied with the new medical scheme is 70%.
- Alternative hypothesis (H1): The proportion of workers satisfied with the new medical scheme is not 70%.
Step 2: Determine the significance level (α):
- The given significant level is 10%, which means α = 0.10.
Step 3: Calculate the test statistic:
We need to calculate the test statistic, which follows the z-distribution for testing proportions. The formula for the test statistic is:
z = (p - P) / √[(P(1-P)) / n]
where:
p = sample proportion
P = hypothesized proportion under the null hypothesis
n = sample size
In this case,
p = 100 / 150 = 0.6667 (66.67%)
P = 0.70 (70%)
n = 150
Substituting the values into the formula:
z = (0.6667 - 0.70) / √[(0.70 * (1 - 0.70)) / 150]
Step 4: Determine the critical value(s):
Since the alternative hypothesis is two-tailed (not equal to), we need to find the critical values for both tails of the distribution.
The critical value for a two-tailed test at a 10% significance level can be found using a z-table. The critical z-value is approximately 1.645.
Step 5: Make a decision:
- If the absolute value of the test statistic (|z|) is greater than the critical value(s), reject the null hypothesis.
- If the absolute value of the test statistic (|z|) is less than or equal to the critical value(s), fail to reject the null hypothesis.
Step 6: Calculate the p-value:
The p-value is the probability of obtaining a test statistic as extreme (or more extreme) than the one observed, assuming the null hypothesis is true.
For a two-tailed test, we calculate the p-value by doubling the area in the tail(s) beyond the absolute value of the test statistic.
Step 7: Compare the p-value with the significance level:
- If the p-value is less than the significance level (p-value < α), reject the null hypothesis.
- If the p-value is greater than or equal to the significance level (p-value ≥ α), fail to reject the null hypothesis.
Now, let's calculate the test statistic and make a decision.
To test the claim that 70% of the workers at ABSA are satisfied with their new medical scheme, we can use a hypothesis test.
Step 1: State the hypotheses.
- Null hypothesis (H0): The proportion of workers satisfied with the new medical scheme is equal to 70%.
- Alternative hypothesis (H1): The proportion of workers satisfied with the new medical scheme is not equal to 70%.
Step 2: Set the significance level.
The significance level, often denoted as α, determines the likelihood of rejecting the null hypothesis when it is true. In this case, the significance level is given as 10%, which means α = 0.1.
Step 3: Compute the test statistic.
We will use the z-test for proportions to compute the test statistic. The formula for the z-test is:
z = (p - P) / sqrt((P * (1 - P)) / n)
Where:
- p is the sample proportion (proportion of workers satisfied in the sample)
- P is the hypothesized proportion (70% or 0.7 in this case)
- n is the sample size (150 workers)
Using the given values, we have:
p = 100 / 150 = 0.6667
P = 0.7
n = 150
Plugging these values into the formula, we get:
z = (0.6667 - 0.7) / sqrt((0.7 * (1 - 0.7)) / 150)
Step 4: Determine the critical value(s).
Since this is a two-tailed test, we need to find the critical value that corresponds to a significance level of 0.1/2 = 0.05 (divided by 2 because it's a two-tailed test).
Using a standard normal distribution table, we find that the critical z-value for a 0.05 significance level is approximately ±1.645.
Step 5: Make a decision.
If the calculated test statistic falls outside the range of the critical values, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Step 6: Calculate the p-value.
The p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the one calculated from the sample, assuming the null hypothesis is true.
In this case, since the alternative hypothesis is two-sided, we need to find the probability in both tails of the standard normal distribution. The p-value is calculated as 2 multiplied by the probability of obtaining a z-value as extreme as the calculated test statistic.
Step 7: Make a conclusion.
- If the p-value is less than the significance level (0.1), we reject the null hypothesis.
- If the p-value is greater than or equal to the significance level, we fail to reject the null hypothesis.
Note: The calculated p-value will depend on the specific test statistic obtained in Step 3.
By following this procedure, you can now test the claim using the given information.