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Complete Question
Employees from Company A and Company B both receive annual bonuses. What information would you need to test the claim that the difference in annual bonuses is greater than $100 at the 0.05 level of significance? Write out the hypothesis and explain the testing procedure.
Solution
The information required to run an hypothesis test to compare the annual bonuses of employees at company A and employees at company B.
- First of, we require a sample of bonuses from each of the two firms, to calculate the mean bonus at each of the companies and the standard deviation.
- We also need the data to satisfy the conditions necessary for an hypothesis test.
the major conditions for an hypothesis test is that
1) The sample must be a random sample extracted from the population, with each variable in the sample independent from one another.
2) The sample should be exrracted from a normal distribution population or approximate a normal distribution, so as to ensure almost normal behaviour from the samples
Then, to do the test,
Step 1
For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.
While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.
For this test, we want to investigate if the difference in annual bonuses for employees of company A and employees of company B is greater than $100.
Hence, the null hypothesis is that there is no sufficient evidence to suggest that the difference in annual bonuses for employees from company A and employees from company B is greater than $100.
And the alternative hypothesis is that there is sufficient evidence to suggest that the difference in annual bonuses for employees from company A and employees from company B is greater than $100.
Mathematically, if the difference in mean annual bonuses for employees from company A and employees from company B is μ
The null hypothesis is represented as
H₀: μ ≤ 100
The alternative hypothesis is represented as
Hₐ: μ > 100
Step 2
The testing procedure for this is a two-sample hypothesis test, Hence, we compute the test statistic in the next step.
Note that we must have had the mean of each of the samples (μ₁ and μ₂) and their standard deviations from the sample of annual bonuses of employees from both firms (s₁ and s₂)
Such that μ = μ₁ - μ₂
Test statistic = (μ - 100)/σₓ
σₓ = √[(s₁²/n₁) + (s₂²/n₂)]
Test statistic =
[(μ₁ - μ₂) - 100] ÷ √[(s₁²/n₁) + (s₂²/n₂)]
Step 3
Using the test statistic calculated, the significance level, the degree of freedom and the p-value table, we obtain the p-value.
Degree of freedom = n₁ + n₂ - 2
Step 4
The interpretation of p-values
When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.
Hope this Helps!!!
