The U.S. Department of Labor reported the average hourly...

The U.S. Department of Labor reported the average hourly earnings for U.S. production workers to be $\$ 14.32$ per hour in 2001 (The World Almanac, 2003 ). A sample of 75 production workers during 2003 showed a sample mean of $\$ 14.68$ per hour. Assuming the population standard deviation $\sigma=\$ 1.45,$ can we conclude that an increase occurred in the mean hourly carnings since $2001 ?$ Use $\alpha=.05$

Key Concepts

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to make inferences about a population parameter based on sample data. It involves setting up a null hypothesis (typically representing the status quo or no effect) and an alternative hypothesis (representing the effect or change of interest), then using sample data to determine if there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis (H0) represents the assumption that there is no change or difference, while the alternative hypothesis (H1 or Ha) represents the claim one seeks to support. In the context of testing for an increase in the mean hourly earnings, the null hypothesis would state that the mean earnings have not increased, whereas the alternative hypothesis would state that they have increased.

Significance Level (?)

The significance level, denoted by ?, is the threshold for determining statistical significance. It represents the probability of rejecting the null hypothesis when it is actually true (a type I error). A common choice is ? = 0.05, which means there is a 5% risk of concluding that an effect exists when it does not.

Z-Test (Test Statistic)

When the population standard deviation is known, as in this scenario, the Z-test is used to compare the sample mean to the hypothesized population mean. The test statistic is calculated by subtracting the hypothesized mean from the sample mean, and then dividing by the standard error, which is the population standard deviation divided by the square root of the sample size. This statistic is then compared against a critical value from the standard normal distribution.

Sampling Distribution and the Central Limit Theorem

The sampling distribution of the sample mean is a probability distribution that describes how the sample mean would vary from sample to sample. The Central Limit Theorem states that, for a sufficiently large sample size (usually n ? 30), this distribution will be approximately normal, regardless of the shape of the population distribution. This property is crucial for justifying the use of the Z-test in hypothesis testing.

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