The mean salary of federal government employees on the...

The mean salary of federal government employees on the General Schedule is $\$ 59,593 .$ The average salary of 30 randomly selected state employees who do similar work is $\$ 58,800$ with $\sigma=\$ 1500$. At the 0.01 level of significance, can it be concluded that state employees earn on average less than federal employees?

Key Concepts

P-value

The p-value is the probability of obtaining results as extreme as those observed, assuming that the null hypothesis is true. It serves as a tool to decide if the observed data is inconsistent with the null hypothesis; if the p-value is less than the chosen significance level, the null hypothesis is rejected.

Test Statistic

The test statistic is a standardized value that quantifies the difference between the sample data and what is expected under the null hypothesis. It is calculated using sample data (such as the sample mean, sample standard deviation, and sample size) and is used to determine the p-value or to compare against critical values from a theoretical distribution.

Standard Error

The standard error measures the variability or dispersion of the sample mean estimate from the true population mean. It is calculated by dividing the sample standard deviation by the square root of the sample size, and it plays a crucial role in determining the precision of the estimated mean in hypothesis testing.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis generally posits no effect or no difference between groups, whereas the alternative hypothesis suggests that there is an effect or a difference. For example, in comparing salaries, the null hypothesis might state that there is no difference in average salaries, while the alternative hypothesizes that one group’s average salary is lower than the other’s.

Hypothesis Testing

This is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a specific condition is true for the entire population. It involves formulating a null hypothesis that represents a baseline or default claim and an alternative hypothesis that represents the claim to be tested, then using sample data to decide which hypothesis is more likely to be true.

Significance Level

The significance level, often denoted by alpha (?), represents the probability of rejecting the null hypothesis when it is actually true (Type I error). It sets a threshold for deciding whether an observed effect is statistically significant, with common choices being 0.05, 0.01, etc.

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