Individuals filing federal income tax returns prior to March...

Individuals filing federal income tax returns prior to March 31 received an average refund of $$\$10$$S6. Consider the population of "last-minute" fllers who mail their tax return during the last five days of the income tax period (typically April 10 to April 15). a. A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. Develop appropriate hypotheses such that rejection of $H_0$ will support the researcher's contention. b. For a sample of 400 individuals who filed a tax return between April 10 and 15 , the sample mean refund was $$\$ 910$$. Based on prior experience a population standard deviation of $$\sigma=\$ 1600$$ may be assumed. What is the $p$-value? c. At $a=.05$, what is your conclusion? d. Repear the preceding hypochesis test using the critical value approach.

Individuals filing federal income tax returns prior to March 31 received an average refund of $$\$10$$S6. Consider the population of "last-minute" fllers who mail their tax return during the last five days of the income tax period (typically April 10 to April 15).
a. A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. Develop appropriate hypotheses such that rejection of $H_0$ will support the researcher's contention.
b. For a sample of 400 individuals who filed a tax return between April 10 and 15 , the sample mean refund was $$\$ 910$$. Based on prior experience a population standard deviation of $$\sigma=\$ 1600$$ may be assumed. What is the $p$-value?
c. At $a=.05$, what is your conclusion?
d. Repear the preceding hypochesis test using the critical value approach.

Key Concepts

Critical Value Approach

This method involves determining the cutoff value (or values) from the statistical distribution (such as the standard normal distribution for a z-test) that correspond to the chosen significance level. If the calculated test statistic exceeds this critical value (or falls in the rejection region), the null hypothesis is rejected. This approach complements the p-value method and leads to the same conclusion when the test is properly conducted.

Significance Level

The significance level (often denoted by alpha, such as 0.05) is the threshold at which the p-value is compared to decide whether to reject the null hypothesis. It represents the probability of committing a Type I error, meaning the risk of rejecting the null hypothesis when it is actually true.

P-Value

The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true. In the context of a hypothesis test, a small p-value indicates that the observed data would be very unlikely under the null hypothesis, thus providing evidence to reject it.

Formulation of Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the claim to be tested, often indicating no change or no difference. The alternative hypothesis (Ha) is what the researcher wants to support, and it is usually directional if there's an expected decrease or increase. Formulating these hypotheses correctly is crucial because the direction of the alternative (e.g., one-tailed vs. two-tailed) determines how the test is conducted and how evidence is accumulated.

One-Sample Z-Test

A one?sample z-test is used when testing hypotheses about the mean of a population when the population standard deviation is known and the sample size is sufficiently large. This test standardizes the difference between the sample mean and the hypothesized population mean into a z-score, allowing the use of the standard normal distribution to determine probabilities.

Hypothesis Testing

This is the fundamental process of making a decision about a population parameter based on sample data. It involves setting up a null hypothesis (usually a statement of no effect or no difference) and an alternative hypothesis (which reflects the researcher's expectation). The hypothesis test evaluates whether the sample data provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

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