10. Hypothesis Testing for Two Samples
Two Means - Unknown, Unequal Variance
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Researchers are comparing the average number of hours worked per week by employees at two different companies. Below are the results from two independent random samples. Assuming population standard deviations are unknown and unequal, calculate the -score for the difference in means, but do not find a -value or state a conclusion.
Company A: ; hours; hours
Company B: hours; hours
A
1.316
B
1.344
C
1.012
D
1.034
Verified step by step guidance1
Step 1: Identify the formula for the t-score when comparing the difference in means for two independent samples with unequal variances. The formula is: t = (xÌ„â‚ - xÌ„â‚‚) / sqrt((s₲ / nâ‚) + (s₂² / nâ‚‚)), where xÌ„â‚ and xÌ„â‚‚ are the sample means, sâ‚ and sâ‚‚ are the sample standard deviations, and nâ‚ and nâ‚‚ are the sample sizes.
Step 2: Substitute the given values into the formula. For Company A: x̄₠= 22.4, s₠= 3.2, n₠= 25. For Company B: x̄₂ = 21.1, s₂ = 2.9, n₂ = 16. The formula becomes: t = (22.4 - 21.1) / sqrt((3.2² / 25) + (2.9² / 16)).
Step 3: Simplify the numerator. Calculate the difference in sample means: (22.4 - 21.1).
Step 4: Simplify the denominator. First, square the standard deviations: s₲ = 3.2² and s₂² = 2.9². Then divide each squared value by its respective sample size: (s₲ / nâ‚) and (s₂² / nâ‚‚). Finally, add these two results together and take the square root.
Step 5: Divide the simplified numerator by the simplified denominator to compute the t-score. This will give you the t-score for the difference in means.
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