Amstat News (December 2004) lists median salaries for...

Amstat News (December 2004) lists median salaries for associate professors of statistics at research institutions and at liberal arts and other institutions in the United States. Assume that a sample of 200 associate professors from research institutions has an average salary of $$\$ 70,750$$ per year with a standard deviation of $$\$ 6000$$. Assume also that a sample of 200 associate professors from other types of institutions has an average salary of $$\$ 65,200$$ with a standard deviation of $$\$ 5000$$. Test the hypothesis that the mean salary for associate professors in research institutions is $\mathbf{8 2 0 0 0}$ higher than for those in other institutions. Use a 0.01 level of significance.

Key Concepts

Central Limit Theorem

The Central Limit Theorem is a foundational statistical concept which states that, for sufficiently large sample sizes, the sampling distribution of the sample mean (or the difference between two sample means) will tend to be approximately normally distributed, regardless of the shape of the population distribution. This justifies using normal probability models (such as Z-tests) when performing inference on sample means with large samples.

Significance Level

The significance level, denoted by alpha (?), is the threshold probability for rejecting the null hypothesis. It represents the risk of falsely rejecting the null hypothesis (a Type I error) and is chosen by the researcher before testing. A significance level of 0.01, for example, indicates that there is a 1% risk of rejecting the null hypothesis when it is actually true.

Test Statistic and Standard Error Calculation

The test statistic is a standardized value calculated from the sample data that measures the distance between the observed sample statistic and the value specified under the null hypothesis, relative to the standard error of the difference. The standard error quantifies the variability of the difference between two sample means, which is essential in determining how extreme the observed result is under the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) is the assumption that there is no difference or no effect, while the alternative hypothesis (Ha) is the assertion that there is a difference or an effect. For testing differences in population means, the null hypothesis typically posits that the difference equals a specific value, and the alternative hypothesis reflects a claim that the difference is not equal to that value, is greater than, or is less than that value.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sampled data. It involves setting up a null hypothesis, which represents a default or no-effect statement, and an alternative hypothesis, which represents the claim being tested. The test procedure computes a test statistic from the sample data to decide whether the null hypothesis can be rejected in favor of the alternative hypothesis.

Two-Sample Inference for Means

Two-sample inference for means involves comparing the means from two independent samples to determine if there is a statistically significant difference between the populations they represent. This concept relies on the assumption that the samples are independent and representative, because the comparison of their means can then be used to infer differences at the population level.

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