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Chapter 10: Problem 16

(a) determine the null and alternative hypotheses, (b) explain what it would mean to make a Type I error, and (c) explain what it would mean to make a Type II error. According to Giving and Volunteering in the United States, 2001 Edition, the mean charitable contribution per household in the United States in 2000 was \(\$ 1623 .\) A researcher believes that the level of giving has changed since then.

Short Answer

Expert verified
The hypothesis test is: \text\bf\bf{\bf鈦�;\sim 1623H_1: 渭 鈮� 1623.\鈦碩he Type I error means incorrectly concluding that the mean has changed, and Type II error means incorrectly concluding it has not changed.

Step by step solution

01

- State the null and alternative hypotheses

The null hypothesis (H_0\textsl{H_0}H_0鈥�) represents the status quo or a statement of no change. The alternative hypothesis (H_A\textsl{H_A}H_A鈥�) represents the researcher's claim. Here, the null hypothesis is that the mean charitable contribution per household in the U.S. in 2000 is \text\bf\bf{\sim 1623}; the alternative hypothesis is that it has changed.\[H_0: 渭 = 1623\]\[H_A: 渭 鈮� 1623\]
02

- Explain Type I error

A Type I error occurs when the null hypothesis is true but is incorrectly rejected. In this context, it would mean concluding that the mean charitable contribution per household has changed when, in fact, it has not.
03

- Explain Type II error

A Type II error occurs when the null hypothesis is false but fails to be rejected. Here, it would mean concluding that the mean charitable contribution per household has not changed since 2000, when, in reality, it has changed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
In hypothesis testing, the null hypothesis (often denoted as \(H_0\)) represents the default or status quo assumption that there is no effect or no change. It is essentially a statement that the researcher tries to disprove. For the provided exercise, the null hypothesis is that the mean charitable contribution per household in the United States in 2000 is still \(\text{\textdollar} 1623\). This is mathematically represented as \[H_0: 渭 = 1623 \] Here, \(渭\) is the population mean. The null hypothesis is crucial because it sets the benchmark against which the alternative hypothesis will be tested.
Alternative Hypothesis
The alternative hypothesis (denoted as \(H_A\)) is what the researcher aims to prove. It contradicts the null hypothesis and indicates a change or effect. In this exercise, the researcher's belief that the level of charitable contributions has changed since 2000 forms the alternative hypothesis. This is expressed as \[H_A: 渭 eq 1623 \] Here, the symbol \( eq \) means 'not equal to,' implying that the mean charitable contribution per household is different from \(\text{\textdollar} 1623\). The alternative hypothesis is vital as it highlights the specific claim the researcher wants to test.
Type I Error
Type I error occurs when the null hypothesis is true but is incorrectly rejected. In other words, it means concluding that there is an effect or change when there actually isn't. For our charitable contribution example, a Type I error would mean concluding that the mean charitable contribution per household has changed since 2000 when, in fact, it has not. Think of it as a 'false positive.' The consequences of a Type I error can be serious because it may lead to unnecessary changes or interventions based on incorrect findings.
Type II Error
A Type II error happens when the null hypothesis is false but fails to be rejected, meaning the test concludes there is no effect when there actually is one. In the context of the charitable contribution example, a Type II error would occur if the test concludes that the mean charitable contribution per household has not changed since 2000, when, in reality, it has changed. This is often called a 'false negative.' Type II errors can be detrimental as they may prevent necessary policy changes or interventions, leading to missed opportunities for improvements.
Charitable Contribution
Charitable contributions refer to the donations made by individuals or households to various causes, such as non-profit organizations, educational institutions, or healthcare facilities. In the given exercise, the average charitable contribution per household in the United States in 2000 was \(\text{\textdollar} 1623\). Researchers may study contributions over time to understand trends, shifts in generosity, or the impact of economic changes. The study of charitable contributions is essential for non-profits and policymakers to tailor strategies to boost giving and ensure resources are effectively utilized.

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Most popular questions from this chapter

Explain what it means to make a Type I error. Explain what it means to make a Type II error.

Conduct the appropriate test. A simple random sample of size \(n=320\) adults was asked their favorite ice cream flavor. Of the 320 individuals surveyed, 58 responded that they preferred mint chocolate chip. Test the claim that less than \(25 \%\) of adults prefer mint chocolate chip ice cream at the \(\alpha=0.01\) level of significance.

Suppose the consequences of making a Type I error are severe. Would you choose the level of significance, \(\alpha,\) to equal \(0.01,0.05,\) or \(0.10 ?\) Why?

Simulation Simulate drawing 40 simple random samples of size \(n=35\) from a population that is exponentially distributed with mean 8 and standard deviation \(\sqrt{8}\). (a) Test the null hypothesis \(H_{0}: \mu=8\) versus the alternative hypothesis \(H_{1}: \mu \neq 8\) (b) Suppose we were testing this hypothesis at the \(\alpha=0.05\) level of significance. How many of the 40 samples would you expect to result in a Type I error? (c) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (b)? (d) Describe how we know that a rejection of the null hypothesis results in making a Type I error in this situation.

Volume of Dell Computer Stock The average daily volume of Dell Computer stock in 2000 was \(\mu=31.8\) million shares, with a standard deviation of \(\sigma=14.8\) million shares, according to Yahoo!Finance. Based on a random sample of 35 trading days in \(2004,\) the sample mean number of shares traded is found to be 23.5 million. Is the volume of Dell stock different in \(2004 ?\) Use the \(\alpha=0.05\) level of significance.
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