Hypothesis Testing Using Rejection Regions In Exercises 23–30, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic X^2, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.


Salaries The annual salaries (in dollars) of 15 randomly chosen senior level graphic design specialists are shown in the table at the left. At α=0.05, is there enough evidence to support the claim that the standard deviation of the annual salaries is different from \$13,056?

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (H0) and the alternative hypothesis (Ha). The claim is that the standard deviation of the annual salaries is different from \$13,056. Thus, H0: σ = \$13,056 (the standard deviation is equal to \$13,056), and Ha: σ ≠ \$13,056 (the standard deviation is different from \$13,056).
2. Step 2: Determine the critical value(s) and rejection region(s). Since this is a two-tailed test (due to the 'different' claim), use the chi-square distribution table with degrees of freedom (df = n - 1, where n is the sample size). The significance level is α = 0.05, so divide α by 2 for each tail (α/2 = 0.025). Look up the critical values for df = 14 in the chi-square table.
3. Step 3: Calculate the standardized test statistic X². Use the formula X² = ((n - 1) * s²) / σ², where n is the sample size, s is the sample standard deviation (calculated from the given data), and σ is the hypothesized standard deviation (\$13,056). First, compute the sample variance (s²) using the salaries provided, then substitute the values into the formula.
4. Step 4: Compare the test statistic X² to the critical values. If X² falls within the rejection region (either below the lower critical value or above the upper critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is enough evidence to support the claim that the standard deviation of the annual salaries is different from \$13,056. If the null hypothesis is not rejected, conclude that there is not enough evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H0. If the calculated test statistic falls within this region, we reject the null hypothesis.

Standard Deviation and Chi-Square Test

The standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of hypothesis testing for standard deviation, the Chi-Square test is used to determine if the sample standard deviation significantly differs from a specified value. This test compares the observed variance in the sample to the expected variance under the null hypothesis, allowing us to assess whether the claim about the population standard deviation is supported by the data.