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Chapter 9: Problem 52

a. Define \(\alpha\) and \(\beta\) for a statistical test of hypothesis. b. For a fixed sample size \(n\), if the value of \(\alpha\) is decreased, what is the effect on \(\beta ?\) c. In order to decrease both \(\alpha\) and \(\beta\) for a particular alternative value of \(\mu\), how must the sample size change?

Short Answer

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Question: Define alpha and beta in statistical hypothesis testing, and explain how to decrease both alpha and beta. Answer: Alpha (伪) is the type I error rate representing the probability of rejecting the null hypothesis (H_0) when it is true. Beta (尾) is the type II error rate representing the probability of not rejecting the null hypothesis (H_0) when it is false. To decrease both alpha and beta, we need to increase the sample size (n), which provides more information about the population and allows for a more accurate estimation of the true parameter, leading to a decrease in both type I and type II error rates.

Step by step solution

01

Part a: Definition of Alpha and Beta in Statistical Hypothesis Test

Alpha, denoted by \(\alpha\), represents the type I error rate, which is the probability of rejecting the null hypothesis \(H_0\) when it is actually true. In other words, it is the false positive rate. Beta, denoted by \(\beta\), represents the type II error rate, which is the probability of not rejecting the null hypothesis \(H_0\) when it is actually false. In other words, it is the false negative rate.
02

Part b: Effect of decreasing alpha on beta

When the value of \(\alpha\) is decreased for a fixed sample size \(n\), it means we are being more conservative in rejecting the null hypothesis. As a result, the chance of false positives (Type I errors) will decrease. However, this may lead to an increase in the probability of not rejecting a false null hypothesis (Type II errors), i.e., \(\beta\) is likely to increase.
03

Part c: Reducing both Alpha and Beta

To decrease both \(\alpha\) and \(\beta\) for a particular alternative value of \(\mu\), we need to increase the sample size \(n\). A larger sample size provides more information about the population and allows for a more accurate estimation of the true parameter. This leads to a decrease in both type I and type II error rates, i.e., \(\alpha\) and \(\beta\).

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Key Concepts

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

Type I Error
The Type I error, represented by the symbol \( \alpha \), refers to the probability of incorrectly rejecting the null hypothesis \( H_0 \) when it is true. Think of it as a false alarm or a false positive. Imagine you're conducting a medical test that indicates a disease is present when it is not. This mistake is a Type I error.

Type I errors are significant because they can lead to incorrect conclusions and unnecessary actions. To control the Type I error rate, researchers predefine \( \alpha \) before the study, often setting it at 0.05 or 5%.

Key points to remember about Type I errors:
  • \( \alpha \) is the threshold level for accepting risk of a false positive.
  • Lowering \( \alpha \) makes you more cautious, reducing false positives but potentially increasing other risks.
  • Choosing the right \( \alpha \) depends on the context and consequences of error.
Type II Error
Type II error, represented by \( \beta \), is the probability of failing to reject the null hypothesis \( H_0 \) when it is actually false. This is known as a false negative. Picture a situation where a test fails to alert you about a real fire hazard. That's a Type II error.

Understanding Type II errors is essential as they reflect the power of a test to detect an effect when one exists. In hypothesis testing, power is calculated as \( 1 - \beta \).

Here are some key insights about Type II errors:
  • The lower the \( \beta \), the higher the test's power and probability of detecting a true effect.
  • Increasing \( \alpha \) usually decreases \( \beta \), but this isn't straightforward as it involves trade-offs.
  • Researchers often aim for a \( \beta \) of 0.2, implying 80% power.
Sample Size
Sample size plays a crucial role in hypothesis testing, impacting both the Type I and Type II error rates. When researchers want to minimize both \( \alpha \) and \( \beta \), increasing the sample size is key.

A larger sample size offers a more precise estimate of the population parameter, which reduces variability and directly influences error rates.

Consider these points about sample size:
  • Increasing sample size can decrease both \( \alpha \) and \( \beta \), improving test accuracy.
  • Larger samples provide more reliable results and more power to detect true effects.
  • Balancing cost and feasibility with desired precision is crucial when determining sample size.
By understanding the relationship between sample size and error rates, you can design more effective studies and make stronger statistical inferences.

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Most popular questions from this chapter

Generation Next Born between 1980 and 1990, Generation Next was the topic of Exercise \(8.64 .^{17}\) In a survey of 500 female and 500 male students in Generation Next, 345 of the females and 365 of the males reported that they decided to attend college in order to make more money. a. Is there a significant difference in the population proportions of female and male students who decided to attend college in order to make more money? Use \(\alpha=.01\) b. Can you think of any reason why a statistically significant difference in these population proportions might be of practical importance? To whom might this difference be important?

A random sample of \(n=1400\) observations from a binomial population produced \(x=529 .\) a. If your research hypothesis is that \(p\) differs from .4, what hypotheses should you test? b. Calculate the test statistic and its \(p\) -value. Use the \(p\) -value to evaluate the statistical significance of the results at the \(1 \%\) level. c. Do the data provide sufficient evidence to indicate that \(p\) is different from \(.4 ?\)

What's Normal? What is normal, when it comes to people's body temperatures? A random sample of 130 human body temperatures, provided by Allen Shoemaker \(^{3}\) in the Journal of Statistical Education, had a mean of \(98.25^{\circ}\) and a standard deviation of \(0.73^{\circ} .\) Does the data indicate that the average body temperature for healthy humans is different from \(98.6^{\circ},\) the usual average temperature cited by physicians and others? Test using both methods given in this section. a. Use the \(p\) -value approach with \(\alpha=.05\). b. Use the critical value approach with \(\alpha=.05\). c. Compare the conclusions from parts a and b. Are they the same? d. The 98.6 standard was derived by a German doctor in 1868 , who claimed to have recorded 1 million temperatures in the course of his research. \({ }^{4}\) What conclusions can you draw about his research in light of your conclusions in parts a and b?

Taste Testing In a head-to-head taste test of store-brand foods versus national brands, Consumer Reports found that it was hard to find a taste difference in the two. \({ }^{10}\) If the national brand is indeed better than the store brand, it should be judged as better more than \(50 \%\) of the time. a. State the null and alternative hypothesis to be tested. Is this a one- or a two-tailed test? b. Suppose that, of the 35 food categories used for the taste test, the national brand was found to be better than the store brand in eight of the taste comparisons. Use this information to test the hypothesis in part a. Use \(\alpha=.01\). What practical conclusions can you draw from the results?

Does College Pay Off? An article in Time describing various aspects of American life indicated that higher educational achievement paid off! College grads work 7.4 hours per day, fewer than those with less than a college education. \({ }^{2}\) Suppose that the average work day for a random sample of \(n=100\) indivic uals who had less than a 4 -year college education was calculated to be \(\bar{x}=7.9\) hours with a standard deviation of \(s=1.9\) hours. a. Use the \(p\) -value approach to test the hypothesis tha the average number of hours worked by individual having less than a college degree is greater than individuals having a college degree. At what level can you reject \(H_{0} ?\) b. If you were a college graduate, how would you state your conclusion to put yourself in the best possible light? c. If you were not a college graduate, how might you state your conclusion?
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