Question

For Exercises 5 through \(20,\) perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. The U.S. median annual income for men in 2014 (in constant dollars) was \(\$ 35,642 .\) A random sample of recent male college graduates indicated the following incomes. At the 0.05 level of significance, test the claim that the median is more than \(\$ 35,642\) $$ \begin{array}{lllll} 35,000 & 37,682 & 39,800 & 32,500 & 30,000 \\ 41,050 & 36,198 & 31,500 & 29,650 & 35,800 \\ 34,500 & 38,850 & 39,750 & & \end{array} $$

Short answer

The test suggests that the median income is significantly greater than $35,642.

Step-by-step solution

Stating the Hypotheses

In hypothesis testing, we start by establishing a null hypothesis and an alternative hypothesis. The null hypothesis (_0]) states the median annual income for recent male college graduates is [35,642]. The alternative hypothesis (_a]) states that the median is more than [35,642]. Therefore, we have: - _0: M = 35,642] - _a: M > 35,642] The claim is represented by the _a].

Finding the Critical Value

The critical value is determined by the chosen significance level, which is [0.05]. Since this is a right-tailed test, we look up the critical value corresponding to [0.05] in a standard normal distribution table or use a t-distribution table depending on the sample size. For our purposes, assuming a large sample approximation, the critical z-value is approximately [1.645].

Computing the Test Value

Calculate the test statistic using the sample data. For a median test with small samples, you could use a sign test to determine the proportion of sample data above the hypothesized median. List incomes relative to [35,642]: - Below median: 35,000, 32,500, 30,000, 31,500, 29,650, 35,800, 34,500 - Above median: 37,682, 39,800, 41,050, 36,198, 38,850, 39,750 This results in 6 values above [35,642] and 7 below. The difference from the median (more or less, simplifying for signs) means the counts can be submitted to a binomial test of significance. Note this test value against critical values.

Making the Decision

Compare the computed test value with the critical value from the z-table or binomial table. Here, we calculate a z-value for this proportion or check if the binomial number of 6 exceeds table limits. If the test value is greater than [1.645] or equivalent binomial validity shows a low probability (p < 0.05), we reject the null hypothesis.

Summarizing the Results

After performing the test, if our calculated test value (or binomial probability) shows that greater than half of the incomes, in this context, signify a statistical reject of the median against alternative hypothesis proportions, that implies recent male college graduates have a median income greater than the hypothesized value, albeit at our significance level. Thus, we affirm _a: M > 35,642], concluding this difference statistically significant.

Key concepts

Critical Value

The critical value is a vital component in hypothesis testing. It is a threshold that helps determine whether you should reject the null hypothesis. When performing a hypothesis test, you choose a significance level, denoted as \(\alpha\), which represents the probability of rejecting a true null hypothesis.

For many tests, like the one in our example with a 0.05 significance level, the critical value corresponds to a standard score from a normal distribution, often referred to as a z-score. In right-tailed tests like ours, where we suspect the true median income might be more than $35,642, the critical value separates the critical region, where we would reject the null hypothesis, from the rest of the distribution.

In our exercise, the critical z-value is approximately 1.645, which means if our test statistic exceeds this value, the results are statistically significant, prompting us to reject the null hypothesis. Understanding critical values helps us relate statistical findings to practical decision-making.

Test Statistic

The test statistic provides a way to summarize sample data into a single figure when testing a hypothesis. In the example given, we aim to determine if the median income for recent male college graduates is significantly greater than $35,642.

For smaller samples, non-parametric methods like the sign test can be used. This involves counting the number of observations above and below the claimed median. In our exercise's dataset, the division of incomes shows 6 values above and 7 below the hypothesized median. We then compute if this split significantly suggests an income higher than the median.

If the calculated test statistic shows an observed pattern unlikely under the null hypothesis (like surpassing our critical value of 1.645), it indicates a statistically significant result worth rejecting \(H_0\). The test statistic thus not only measures the central tendency but also guides inference.

Median Income

The median income is an essential measure in statistics representing the middle value of a data set when arranged in order. It is more robust than the mean as a measure of central tendency, especially when a dataset includes outliers or is skewed.

In our exercise, the goal is to test if the median income of recent male college graduates is greater than $35,642. Hypotheses are set based on this baseline, and sample data can confirm or refute this through statistical testing.

When dealing with income data, the median is often preferred over the mean because it provides a better picture of the typical income by minimizing the effects of extremely high or low values. This perspective helps in gaining insights into economic conditions and demographic comparisons.

Significance Level

The significance level, denoted as \(\alpha\), is a critical concept in hypothesis testing that indicates the threshold for deciding when to reject a null hypothesis. In practice, it represents the risk you are willing to take to make a Type I error: rejecting a true null hypothesis.

Common significance levels are 0.05, 0.01, and 0.10. In the current exercise, a 0.05 level means a 5% risk of mistakingly rejecting the null hypothesis that the median income is $35,642. It thus provides a balance between the risk of errors and the strictness of decision making.

Choosing a significance level is a subjective decision that can affect the outcomes of your hypothesis test and any conclusions drawn. A lower significance level would demand stronger evidence to reject the null hypothesis, while a higher one would be more permissive. Adjusting this level impacts our confidence in statistical decisions and reflects on the reliability of claims made based on sample data.