Testing a Difference Other Than Zero Sometimes a researcher is interested in testing a difference in means other than zero. In Exercises 27 and 28, you will test the difference between two means using a null hypothesis of Ho: μ1-μ2=k, Ho: μ1-μ2>=k or Ho: μ1-μ2<=k . The standardized test statistic is still
Software Engineer Salaries Is the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, Connecticut, more than \$4000? To decide, you select a random sample of entry level software engineers from each city. The results of each survey are shown in the figure at the left. Assume the population standard deviations are σ1=\$14,060 and σ2=\$13,050 . At α=0.05, what should you conclude? (Adapted from Salary.com)

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). In this case, H₀: μ₁ - μ₂ ≤ 4000 (the difference in mean salaries is less than or equal to \$4000), and Hₐ: μ₁ - μ₂ > 4000 (the difference in mean salaries is greater than \$4000).
2. Step 2: Identify the given data from the problem. For Santa Clara, CA: sample mean (x̄₁) = \$88,900, sample size (n₁) = 42, population standard deviation (σ₁) = \$14,060. For Greenwich, CT: sample mean (x̄₂) = \$81,600, sample size (n₂) = 38, population standard deviation (σ₂) = \$13,050.
3. Step 3: Calculate the standard error (SE) of the difference in means using the formula: SE = √((σ₁² / n₁) + (σ₂² / n₂)). Substitute the values for σ₁, σ₂, n₁, and n₂ into the formula.
4. Step 4: Compute the test statistic (z) using the formula: z = ((x̄₁ - x̄₂) - k) / SE, where k = 4000 (the hypothesized difference). Substitute the values for x̄₁, x̄₂, k, and SE into the formula.
5. Step 5: Compare the calculated z-value to the critical z-value for α = 0.05 in a one-tailed test. If the calculated z-value is greater than the critical z-value, reject the null hypothesis (H₀). Otherwise, fail to reject H₀.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0. In this case, the null hypothesis tests if the difference in means between two groups is equal to a specified value (k), which is crucial for assessing salary differences.

Standardized Test Statistic

A standardized test statistic is a numerical value that measures how far a sample statistic is from the null hypothesis, expressed in terms of standard deviations. It allows for comparison across different datasets and is essential for determining the significance of the results. In this scenario, the test statistic will help evaluate whether the observed difference in salaries exceeds the hypothesized difference of $4,000.

Significance Level (α)

The significance level (α) is the threshold for determining whether to reject the null hypothesis, commonly set at 0.05. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this analysis, using α=0.05 means that if the p-value of the test statistic is less than 0.05, the difference in salaries will be considered statistically significant.