According to the local union president, the mean income of...

According to the local union president, the mean income of plumbers in the Salt Lake City follows the normal distribution. This normal distribution has a mean of $\$ 45,000$ and a standard deviation of $\$ 3,000$. A recent investigative reporter for KYAK TV found, for a sample of 120 plumbers, the mean gross income was $\$ 45,500$. At the .10 significance level, is it reasonable to conclude that the mean income is not equal to $\$ 45,000$ ? Determine the $p$ -value.

Key Concepts

Hypothesis Testing

This concept involves forming a null hypothesis and an alternative hypothesis to make a statistical decision about a population parameter based on sample data. In the context of the question, the null hypothesis states that the true mean income is $45,000, while the alternative hypothesis states it is not $45,000. Hypothesis testing provides a framework to decide if observed sample evidence is strong enough to reject the null hypothesis at a predetermined significance level.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, where the mean, median, and mode are equal. In the problem, both the underlying income distribution and the sampling distribution of the sample mean are assumed to be normal, which allows for the application of standard z-test methods to evaluate the hypothesis about the mean.

Z-test

A z-test is a statistical method used when the population standard deviation is known, or the sample size is large enough for the Central Limit Theorem to ensure that the sampling distribution of the mean is approximately normal. In this scenario, the z-test is used to compute the test statistic by comparing the sample mean to the population mean, which then helps in determining the p-value.

p-value

The p-value represents the probability of obtaining a test statistic as extreme as the one observed, or more extreme, under the assumption that the null hypothesis is true. It quantifies the evidence against the null hypothesis. In the given problem, a low p-value relative to the significance level would indicate that the sample data provide sufficient evidence to conclude that the true mean income differs from $45,000.

Significance Level

The significance level, often denoted as alpha (?), is the threshold used to decide whether the observed data is statistically significant. It defines the probability of rejecting the null hypothesis when it is actually true (Type I error). In the question, a significance level of 0.10 is used, meaning that if the p-value is less than 0.10, the null hypothesis would be rejected in favor of the alternative hypothesis.

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