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: x1 = 648, s12 = 120, and n1 = 100
District Y: x2 = 495, s22 = 140, and n2 = 90
Verify at 0.05 level of significance whether the sample results support the claim of the organization.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Is it more or less than .05?
To verify whether the sample results support the claim of the organization, we can use a two-sample t-test. Here are the step-by-step instructions to perform the test:
Step 1: State the null hypothesis (H0) and the alternative hypothesis (H1):
- Null hypothesis (H0): The monthly wages of industrial workers in district X do not exceed that of those in district Y by more than Rs 150. (µ1 - µ2 ≤ 150)
- Alternative hypothesis (H1): The monthly wages of industrial workers in district X exceed that of those in district Y by more than Rs 150. (µ1 - µ2 > 150)
Step 2: Set the significance level (α):
The significance level (α) is given as 0.05, which corresponds to a confidence level of 95%.
Step 3: Compute the test statistic:
The test statistic for a two-sample t-test is given by:
t = (x1 - x2 - d) / sqrt((s1^2 / n1) + (s2^2 / n2))
Where:
- x1 and x2 are the means of the two samples.
- s1 and s2 are the standard deviations of the two samples.
- n1 and n2 are the sizes of the two samples.
- d is the difference under the null hypothesis (150 in this case).
Step 4: Determine the critical value:
To determine the critical value, we need to find the degrees of freedom (df) and consult the t-distribution table. The degrees of freedom can be calculated using the following formula:
df = (s1^2 / n1 + s2^2 / n2)^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
Step 5: Compare the test statistic and critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Let's compute the necessary values and perform the test:
x1 = 648, s1^2 = 120, n1 = 100
x2 = 495, s2^2 = 140, n2 = 90
d = 150 (from the claim)
α = 0.05 (given)
Calculating the test statistic:
t = (x1 - x2 - d) / sqrt((s1^2 / n1) + (s2^2 / n2))
Calculating degrees of freedom:
df = (s1^2 / n1 + s2^2 / n2)^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
Comparing t-value with critical value:
If the t-value > t-critical, reject the null hypothesis. Else, fail to reject the null hypothesis.
I will calculate the test statistic and degrees of freedom for you. Give me a moment.