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Chapter 8: Problem 5

Consider a test for \(\mu\). If the \(P\) -value is such that you can reject \(H_{0}\) for \(\alpha=0.01\), can you always reject \(H_{0}\) for \(\alpha=0.05\) ? Explain.

Short Answer

Expert verified
Yes, rejecting \(H_0\) at \(\alpha = 0.01\) means you can also reject it at \(\alpha = 0.05\).

Step by step solution

01

Understanding the Problem

The question asks about the relationship between rejecting the null hypothesis \(H_0\) at different significance levels, \(\alpha\). Specifically, if \(H_0\) is rejected at \(\alpha = 0.01\), will it always be rejected at \(\alpha = 0.05\)? We need to understand what this means in terms of \(P\)-values and significance levels.
02

Definition of \(P\)-value and \(\alpha\)

The \(P\)-value is the probability of observing the test results under the null hypothesis. A smaller \(P\)-value indicates stronger evidence against \(H_0\). The significance level \(\alpha\) is a threshold for rejecting \(H_0\); we reject \(H_0\) if the \(P\)-value is less than \(\alpha\). Lower \(\alpha\) means more convincing evidence is needed to reject \(H_0\).
03

Comparing \(\alpha = 0.01\) and \(\alpha = 0.05\)

The exercise mentions two significance levels, \(\alpha = 0.01\) and \(\alpha = 0.05\). \(\alpha = 0.01\) requires stronger evidence against \(H_0\) compared to \(\alpha = 0.05\). Hence, if the \(P\)-value is lower than 0.01, it definitely will be lower than 0.05 as well.
04

Conclusion

If you reject \(H_0\) for \(\alpha = 0.01\), it means \(P\)-value < 0.01. Since 0.01 is smaller than 0.05, we can conclude that the same \(P\)-value will also be less than 0.05. Therefore, you can also reject \(H_0\) for \(\alpha = 0.05\).

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Key Concepts

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

Understanding Significance Level
In hypothesis testing, the significance level, denoted by \( \alpha \), plays a crucial role in decision-making. It represents the threshold beyond which we reject the null hypothesis \( H_0 \). A common way to think of \( \alpha \) is as the probability of making a Type I error鈥攊ncorrectly rejecting the true \( H_0 \).

Setting the significance level is like setting a bar. The lower the \( \alpha \), the more stringent the test becomes. For example:
  • \( \alpha = 0.05 \): This means there is a 5% risk of making a Type I error and is often used as a standard threshold in social sciences.
  • \( \alpha = 0.01 \): Here, a 1% risk is allowed, indicating the need for stronger evidence to reject \( H_0 \).
If you need greater confidence before rejecting \( H_0 \), you would choose a lower \( \alpha \). This makes the test more conservative but decreases the risk of a false positive.
Decoding the P-value
The \(P\)-value is a key element in hypothesis testing, as it measures the strength of evidence against the null hypothesis \( H_0 \). Imagine it as a numerical representation of surprise. A low \(P\)-value signals strong evidence against \( H_0 \).

Here's how it works:
  • If your \(P\)-value is less than \(\alpha\), you reject \( H_0 \) because this rare result suggests \( H_0 \) may be false.
  • If the \(P\)-value is greater than \(\alpha\), you do not reject \( H_0 \), as there isn't enough evidence.
Given that a \(P\)-value is a probability, it ranges from 0 to 1. A \(P\)-value less than 0.01 indicates stronger against the null hypothesis compared to a \(P\)-value between 0.01 and 0.05, providing you have set your significance level accordingly as in the given example.
The Concept of Null Hypothesis
The null hypothesis, symbolized as \( H_0 \), is a foundational component of hypothesis testing. It often represents the status quo or a statement of no effect, no difference, or no change.

For instance, in a study on a new drug, \( H_0 \) might propose that the drug has no effect compared to the current treatment. The purpose of the hypothesis test is to evaluate whether there's enough evidence to reject this claim.

Steps to engage with \( H_0 \):
  • Formulate \( H_0 \) as a neutral statement鈥攊t typically involves "equals" sign, like \( \mu = \mu_0 \).
  • Conduct an experiment or study to collect data.
  • Use statistical measures like the \(P\)-value and compare it to your significance level \( \alpha \).
  • If the evidence suggests otherwise, you may "reject \( H_0 \)." However, failing to reject \( H_0 \) doesn't prove it true; it merely indicates insufficient evidence to deem it false.
The decision to reject or not reject \( H_0 \) has crucial implications for drawing conclusions in research and making informed decisions.

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Most popular questions from this chapter

In general, if sample data are such that the null hypothesis is rejected at the \(\alpha=1 \%\) level of significance based on a two-tailed test, is \(H_{0}\) also rejected at the \(\alpha=1 \%\) level of significance for a corresponding one-tailed test? Explain.

Please read the Focus Problem at the beginning of this chapter. Recall that Benford's Law claims that numbers chosen from very large data files tend to have " \(1 "\) as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with " 1 " as the leading digit is about \(0.301\) (see the reference in this chapter's Focus Problem). Now suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of \(n=215\) numerical entries from the file and \(r=46\) of the entries had a first nonzero digit of \(1 .\) Let \(p\) represent the population proportion of all numbers in the corporate file that have a first nonzero digit of \(1 .\) j. Test the claim that \(p\) is less than \(0.301\). Use \(\alpha=0.01\). ii. If \(p\) is in fact less than \(0.301\), would it make you suspect that there are not enough numbers in the data file with leading 1 's? Could this indicate that the books have been "cooked" by "pumping up" or inflating the numbers? Comment from the viewpoint of a stockholder. Comment from the perspective of the Federal Bureau of Investigation as it looks for money laundering in the form of false profits. iii. Comment on the following statement: "If we reject the null hypothesis at level of significance \(\alpha\), we have not proved \(H_{0}\) to be false. We can say that the probability is \(\alpha\) that we made a mistake in rejecting \(H_{0} . "\) Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of "good," socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the \(\mathrm{P} / \mathrm{E}\), or. price-to-earnings, ratio. High \(\mathrm{P} / \mathrm{E}\) ratios may indicate a stock is overpriced. For the S\&P stock index of all major stocks, the mean \(\mathrm{P} / \mathrm{E}\) ratio is \(\mu=19.4\). A random sample of 36 "socially conscious" stocks gave a \(\mathrm{P} / \mathrm{E}\) ratio sample mean of \(\bar{x}=17.9\), with sample standard deviation \(s=5.2\) (Reference: Morningstar, a financial analysis company in Chicago). Does this indicate that the mean \(\mathrm{P} / \mathrm{E}\) ratio of all socially conscious stocks is different (either way) from the mean \(\mathrm{P} / \mathrm{E}\) ratio of the \(S \& P\) stock index? Use \(\alpha=0.05\).

Do you prefer paintings in which the people are fully clothed? This question was asked by a professional survey group on behalf of the National Arts Society (see reference in Problem 30). A random sample of \(n_{1}=59\) people who are conservative voters showed that \(r_{1}=45\) said yes. Another random sample of \(n_{2}=62\) people who are liberal voters showed that \(r_{2}=36\) said yes. Does this indicate that the population proportion of conservative voters who prefer art with fully clothed people is higher than that of liberal voters? Use \(\alpha=0.05\).

This problem is based on information regarding productivity in leading Silicon Valley companies (see reference in Problem 25). In large corporations, an "intimidator" is an employee who tries to stop communication, sometimes sabotages others, and, above all, likes to listen to him- or herself talk. Let \(x_{1}\) be a random variable representing productive hours per week lost by peer employees of an intimidator. \(\begin{array}{llllllll}x_{1}: & 8 & 3 & 6 & 2 & 2 & 5 & 2\end{array}\) A "stressor" is an employee with a hot temper that leads to unproductive tantrums in corporate society. Let \(x_{2}\) be a random variable representing productive hours per week lost by peer employees of a stressor. \(\begin{array}{lllllllll}x_{2}: & 3 & 3 & 10 & 7 & 6 & 2 & 5 & 8\end{array}\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x}_{1}=4.00, s_{1}=2.38, \bar{x}_{2}=5.5\), and \(s_{2}=2.78 .\) ii. Assuming the variables \(x_{1}\) and \(x_{2}\) are independent, do the data indicate that the population mean time lost due to stressors is greater than the population mean time lost due to intimidators? Use a \(5 \%\) level of significance. (Assume the population distributions of time lost due to intimidators and time lost due to stressors are each mound-shaped and symmetric.)
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