According to the local union president, the mean gross...

According to the local union president, the mean gross income of plumbers in the Salt Lake City area follows the normal probability distribution with 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

Sampling Distribution

The sampling distribution of the sample mean is the probability distribution of the mean values obtained from an infinite number of samples of a given size drawn from a population. When the population is normally distributed or the sample size is large, this distribution is approximately normal, which justifies the use of z-scores in performing hypothesis tests.

Two-Tailed Test

A two-tailed test is a hypothesis test where deviations on both sides of the hypothesized parameter value are considered. This approach is used when the alternative hypothesis is non-directional, meaning that evidence is sought for the parameter being either less than or greater than the specified value, thus requiring the p-value to account for extreme values in both tails of the distribution.

Significance Level

The significance level, denoted by alpha (?), is the pre-determined threshold for rejecting the null hypothesis. It represents the maximum acceptable probability of committing a Type I error—incorrectly rejecting a true null hypothesis. The p-value is compared against this level to decide whether to reject the null hypothesis.

P-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining test results at least as extreme as the observed data. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence that the null hypothesis may be false.

Standard Error

The standard error measures the variability of the sample mean from sample to sample and is calculated as the population standard deviation divided by the square root of the sample size. It plays a crucial role in standardizing the difference between the sample mean and the hypothesized population mean, enabling the use of the normal distribution in hypothesis testing.

Test Statistic

The test statistic is a standardized value calculated from sample data that measures the degree of divergence from the null hypothesis. For tests concerning a known standard deviation and normally distributed data, the z-test statistic is computed as (sample mean minus the population mean) divided by the standard error, which quantifies how many standard errors the observed sample mean is away from the hypothesized mean.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents a default or status quo assumption about a population parameter—in this case, that the true mean income equals a specified value—while the alternative hypothesis (H?) represents a claim that contradicts the null, such as the true mean being different. These hypotheses establish the framework for determining whether observed sample data provides sufficient evidence to reject the null hypothesis.

Transcript

00:01 All right, so we've got a union president looking at the mean income of plumbers in a city.

00:08 And they found that this follows a normal probability distribution.

00:13 But the mean of 45 ,000 given standard of deviation.

00:18 So these are the population, the population of plumbers in the city.

00:23 And an investigator reporter found that out of 120 sample plumbers, their mean was a little bit more.

00:31 So the question is, at the point one significance level, is it reasonable to conclude that the mean income is not equal to 40? so because we know the population, we're going to use the z scores, not the t values.

00:50 So we also want to determine the p value.

00:53 So we use the z score.

00:55 So the sample mean is 45 ,500.

01:03 And we're going to subtract off the given population.

01:06 Mean, divided by this population standard deviation, divided by the square root to the sample size, which is 120.

01:17 So there's the z value that we have.

01:20 Oh, let's do our hypotheses.

01:22 We want to say that the assumption is that the mean is 45 ,000.

01:29 But the alternative hypothesis is that no, it's not equal to 45 ,000.

01:40 Now our decision, so we want to split this up, between the two sides of the distribution.

01:45 So we're actually doing 0 .05.

01:50 So to go to a z table.