A research organization claims that the monthly wages of industrial workers in district X
exceeds that of those in district Y by more than Rs 150. Two different samples drawn
independently from the two district yielded the following results:
District X: 1 x = 648, 2
1 s
= 120, and 1 n = 100
District Y: 2 x = 495, 2
2 s = 140, and 2 n = 90
Verify at 0.05 level of significance whether the sample results support the claim of the
organization.
Learn Hypothesis Step by Step
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To verify whether the sample results support the claim of the organization, we can perform a hypothesis test.
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: The mean monthly wages of industrial workers in district X is equal to or less than the mean monthly wages of industrial workers in district Y (μ1 ≤ μ2).
Ha: The mean monthly wages of industrial workers in district X exceeds the mean monthly wages of industrial workers in district Y by more than Rs 150 (μ1 > μ2 + 150).
Step 2: Set the significance level (α):
In this case, the significance level is given as 0.05, which means we have a 5% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 3: Determine the test statistic:
Since the sample sizes are large enough (n1 = 100, n2 = 90), we can use the standard normal distribution for hypothesis testing. The test statistic is calculated as:
z = (x1 - x2 - d) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and d is the claimed difference.
Step 4: Determine the rejection region:
Calculate the critical value from the standard normal distribution for the given significance level (α = 0.05). Since the alternative hypothesis is one-sided (μ1 > μ2 + 150), we are looking for the critical value that determines the rejection region in the upper tail of the distribution.
Step 5: Calculate the test statistic and compare it with the critical value:
Plug in the given values and calculate the test statistic. Compare it with the critical value obtained in Step 4. If the test statistic falls in the rejection region, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Step 6: Interpret the results:
Based on the analysis, we can either conclude that the sample results support the claim of the organization or fail to support it.
Note: Since the formula for the test statistic involves sample means and sample standard deviations, ensure you have taken the square root of the sample variances to calculate the standard deviations.