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Chapter 9: Problem 46

Example 8 tested a therapy for anorexia, using hypotheses \(\mathrm{H}_{0}: \mu=0\) and \(\mathrm{H}_{a}: \mu \neq 0\) about the population mean weight change \(\mu .\) In the words of that example, what would be (a) a Type I error and (b) a Type II error?

Short Answer

Expert verified
Type I error: Concluding therapy has an effect when it doesn't. Type II error: Concluding therapy doesn't have an effect when it does.

Step by step solution

01

Understanding the Hypotheses

The null hypothesis \( \mathrm{H}_{0}: \mu = 0\) suggests that the population mean weight change is zero, indicating no effect of the therapy. The alternative hypothesis \( \mathrm{H}_{a}: \mu eq 0\) suggests that the population mean weight change is non-zero, indicating an effect of the therapy.
02

Defining Type I Error

A Type I error occurs when we reject the null hypothesis \( \mathrm{H}_{0} \) when it is actually true. In this context, it means concluding that there is an effect of the therapy (the population mean weight change is different from zero) when, in reality, there is no effect.
03

Defining Type II Error

A Type II error occurs when we fail to reject the null hypothesis \( \mathrm{H}_{0} \) when it is false. In this context, it means concluding that there is no effect of the therapy (the population mean weight change is zero) when, in fact, there is an effect (the population mean weight change is not zero).

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Key Concepts

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

Type I Error
Let's start by demystifying the Type I error. Imagine you're baking a cake and decide to test a new baking powder, suspecting it could make cakes rise more. You think your wonderful cake experiment shows the baking powder works better.
But, what if you threw out the original baking powder recipe in favor of the new one only to later realize everything was just luck? If the original really was as good or better, you've made a Type I error.
  • In statistics, a Type I error is about saying something exists (like an effect or a difference) when it actually does not.
  • We "reject the null hypothesis" (\(\mathrm{H}_{0}\)) when it is true.
In the anorexia therapy example, if you claim your therapy causes weight change when it doesn鈥檛, you've committed a Type I error.
This is often considered a "false positive,鈥� much like seeing a ghost that isn鈥檛 there.
Type II Error
Now let鈥檚 move on to the Type II error, which is like the opposite twin of Type I error. Suppose you鈥檙e testing for an exciting new cola flavor. People love it, but you're too careful and don't notice the difference in taste.
You continue believing both colas taste the same, even with a distinct new taste. Here, you've made a Type II error.
  • A Type II error happens when you fail to reject the null hypothesis (\(\mathrm{H}_{0}\)) even though it鈥檚 not true.
  • It鈥檚 like being blind to a real effect that indeed exists.
In our therapy scenario, if there truly is a weight change due to treatment, but you proclaim 鈥榥o effect,' you鈥檝e encountered a Type II error.
This usually feels like a "false negative," missing a fact that is potentially "right under your nose."
Null Hypothesis
Finally, what about the null hypothesis itself? Think of it as the starting assumption or the default.
It's like saying, "Nothing exciting is happening here until proven otherwise."
  • The null hypothesis, denoted by \(\mathrm{H}_{0}\), is a statement of no effect or no difference.
  • We always begin our statistical tests with the null hypothesis as our backdrop.
In our weight change therapy example, the null hypothesis \(\mathrm{H}_{0}: \mu = 0\) suggests zero mean change 鈥� implying the therapy has no impact.
It's our job to gather evidence (data) to either overturn this default stance and propose the alternative hypothesis, or stick with it if the evidence is thin.
Think of the null hypothesis as the skeptical friend who says, 鈥淧rove it to me!鈥�

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Most popular questions from this chapter

A significance test about a mean is conducted using a significance level of \(0.05 .\) The test statistic equals \(10.52 .\) The \(\mathrm{P}\) -value is \(0.003 .\) a. If \(\mathrm{H}_{0}\) was true, for what probability of a Type I error was the test designed? b. If the P-value was 0.3 and the test resulted in a decision error, what type of error was it?

Two researchers conduct separate studies to test \(\mathrm{H}_{0}: p=0.50\) against \(\mathrm{H}_{a}: p \neq 0.50,\) each with \(n=400\) a. Researcher A gets 220 observations in the category of interest, and \(\hat{p}=220 / 400=0.550\) and test statistic \(z=2.00 .\) Show that the P-value \(=0.046\) for Researcher A's analysis. b. Researcher \(\mathrm{B}\) gets 219 in the category of interest, and \(\hat{p}=219 / 400=0.5475\) and test statistic \(z=1.90 .\) Show that the P-value \(=0.057\) for Researcher B's analysis. c. Using \(\alpha=0.05,\) indicate in each case from part a and part b whether the result is "statistically significant." Interpret. d. From part a, part b, and part c, explain why important information is lost by reporting the result of a test as "P-value \(\leq 0.05\) " versus "P-value \(>0.05\)," or as "reject \(\mathrm{H}_{0}\) " versus "do not reject \(\mathrm{H}_{0}\)," instead of reporting the actual P-value. e. Show that the \(95 \%\) confidence interval for \(p\) is (0.501,0.599) for Researcher \(\mathrm{A}\) and (0.499,0.596) for Researcher B. Explain how this method shows that, in practical terms, the two studies had very similar results.

For a test of \(\mathrm{H}_{0}: p=0.50,\) the \(z\) test statistic equals 1.04 a. Find the P-value for \(\mathrm{H}_{a}: p>0.50\). b. Find the P-value for \(\mathrm{H}_{a}: p \neq 0.50\). c. Find the P-value for \(\mathrm{H}_{a}: p<0.50 .\) (Hint: The P-values for the two possible one-sided tests must sum to \(1 .)\) d. Do any of the P-values in part a, part b, or part c give strong evidence against \(\mathrm{H}_{0}\) ? Explain.

Examples of hypotheses Give an example of a null hypothesis and an alternative hypothesis about a (a) population proportion and (b) population mean.

Examples \(1,3,\) and 5 referred to a study about astrology. Another part of the study used the following experiment: Professional astrologers prepared horoscopes for 83 adults. Each adult was shown three horoscopes, one of which was the one an astrologer prepared for him or her and the other two were randomly chosen from ones prepared for other subjects in the study. Each adult had to guess which of the three was his or hers. Of the 83 subjects, 28 guessed correctly. a. Define the parameter of interest and set up the hypotheses to test that the probability of a correct prediction is \(1 / 3\) against the astrologers' claim that it exceeds \(1 / 3 .\) b. Show that the sample proportion \(=0.337,\) the standard error of the sample proportion for the test is \(0.052,\) and the test statistic is \(z=0.08\). c. Find the P-value. Would you conclude that people are more likely to select their horoscope than if they were randomly guessing, or are results consistent with random guessing?
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