A 10-year-old survey found that the average net income for self-employed accountants was $74,914. A researcher wants to determine whether this figure has changed( can be changed in both directions) and takes a random sample of 112 self-employed accountants, producing a sample mean of $78,695. Assume the population standard deviation of net incomes for self-employed accountants is $14,530. Is there any evidence to suggest that the average net income has changed at a 5% level of significance?

The Correct null and alternative hypotheses should be

Ho : µ = $74194 and Ha : µ
≠
$74194

Question

A 10-year-old survey found that the average net income for self-employed accountants was $74,914. A researcher wants to determine whether this figure has changed( can be changed in both directions)  and takes a random sample of 112 self-employed accountants, producing a sample mean of $78,695. Assume the population standard deviation of net incomes for self-employed accountants is $14,530. Is there any evidence to suggest that the average net income has changed at a 5% level of significance?

The Correct null and alternative hypotheses should be

Ho : µ = $74194 and Ha : µ 
≠
$74194

BlackTom AI · Accepted Answer

The question asks us to evaluate whether the statement about the correct null and alternative hypotheses is true or false given the problem setup.

Option statement to evaluate: Ho : μ = $74194 and Ha : μ ≠ $74194.

First, recall the problem data: the historical average net income is reported as $74,914. The standard test for a mean with known population standard deviation uses Ho: μ = μ0 and Ha: μ ≠ μ0, where μ0 is the historical average being tested against. Therefore, the correct μ0 should be 74,914, not 74,194.

Now, assessing the accuracy of the provided hypotheses:
- The null hypothesis should specify μ = 74,914, since that is the value given as the historical average in the problem. Using 74,194 is a misprint or incorrect value.
- The alternative hypothesis should reflect a two-sided test μ ≠ 74,914, since the researcher is testing whether the average could have changed in either direction.

Because the number in the proposed Ho is 74,194 rather than 74,914, the statement as written is not correct. This makes the assertion false.

If one were to correct the number to 74,914, the rationale would be: we would set Ho: μ = 74,914 and Ha: μ ≠ 74,914, and then compute the z statistic using σ = 14,530 and n = 112 to determine if the observed sample mean of 78,695 provides evidence against Ho at the 5% level. However, with the incorrect 74,194 value, the stated hypotheses do not match the problem’s given historical mean, rendering the statement inaccurate.

In summary, the given statement uses an incorrect numeric value for the null hypothesis mean, so the statement is false.

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