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Chapter 9: Problem 4

Consider a test for \(\mu\). If the \(P\) -value is such that you can reject \(H_{0}\) at the \(5 \%\) level of significance, can you always reject \(H_{0}\) at the \(1 \%\) level of significance? Explain.

Short Answer

Expert verified
No, because the 1% level requires a smaller \( P \)-value for rejection.

Step by step solution

01

Understanding Significance Levels

The significance levels, often denoted as alpha (\( \alpha \)), are thresholds for deciding whether to reject the null hypothesis \( H_{0} \). A 5% significance level implies that there is a 5% chance of rejecting \( H_{0} \) when it's actually true, while a 1% level implies a 1% chance. Lower significance levels require stronger evidence to reject \( H_{0} \).
02

Interpreting P-values

A \( P \)-value measures the probability of observing a test statistic as extreme as the sample result, assuming \( H_{0} \) is true. If the \( P \)-value is less than or equal to the significance level, \( H_{0} \) is rejected. For example, a \( P \)-value of 0.04 allows rejection at the 5% level (0.04 < 0.05), but not at the 1% level (0.04 > 0.01).
03

Decision Making at Different Levels

If you reject \( H_{0} \) at the 5% level, it means the \( P \)-value is less than 0.05. However, for the 1% level, \( H_{0} \) can only be rejected if the \( P \)-value is less than 0.01. Therefore, a result that leads to rejection at the 5% level may not necessarily lead to rejection at the 1% level.
04

Conclusion

Because the 1% level is more stringent, rejecting \( H_{0} \) at the 5% significance level does not guarantee that \( H_{0} \) can be rejected at the 1% level unless the \( P \)-value is also less than or equal to 0.01.

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Key Concepts

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

p-value interpretation
A p-value is a vital concept in statistical hypothesis testing as it helps determine the strength of the evidence against the null hypothesis, denoted as \( H_0 \). Imagine running an experiment and getting a certain output. The p-value tells you how likely it is to obtain this result if \( H_0 \) is true.
The smaller the p-value, the stronger the evidence against the null hypothesis. For example:
  • A p-value of 0.04 suggests a 4% chance of observing the result, assuming \( H_0 \) is true.
  • This means there鈥檚 a 4% probability that your results are due to random chance.
A crucial takeaway is that if the p-value is less than a predetermined significance level, you might consider \( H_0 \) to be false. However, if the p-value falls above this threshold, the evidence isn鈥檛 strong enough to reject \( H_0 \).
It鈥檚 important to note that the p-value does not tell you the probability that \( H_0 \) is true outright鈥攊t only reflects the probability of your data assuming \( H_0 \) is correct.
significance levels
Significance levels in hypothesis testing act as benchmarks for determining the strength of evidence needed to reject the null hypothesis \( H_0 \). These thresholds, expressed as percentages, include common levels like 5% (0.05) and 1% (0.01).
  • The 5% significance level indicates a 5% risk of making a Type I error鈥攔ejecting a true \( H_0 \).
  • A 1% level means there's only a 1% risk of such an error, demanding more substantial evidence to reject \( H_0 \).
These levels help different fields set their own standards for acceptable evidence strength. In practice:
  • Lower significance levels imply stricter criteria, requiring more compelling evidence for rejection of \( H_0 \).
  • In a judicial analogy, it's like increasing the burden of proof in a court case.
Choosing a proper significance level is vital and often depends on the context or the implications of making an incorrect decision.
null hypothesis rejection criteria
Rejection criteria for the null hypothesis \( H_0 \) are set by comparing the calculated p-value to the predetermined significance level. This determines whether the observed data provide enough evidence against \( H_0 \).
  • If the p-value is less than or equal to the significance level, \( H_0 \) is typically rejected.
  • If the p-value exceeds the significance level, you fail to reject \( H_0 \).
    • To view it practically:
    • A p-value of 0.02 allows rejection of \( H_0 \) at the 5% level (\( 0.02 < 0.05 \)), but not at the 1% level (\( 0.02 > 0.01 \)).
    This is crucial, as a rejection at a higher significance level, like 5%, doesn't assure rejection at a more stringent level, like 1%. Each level sets its own criteria for evidence strength.
    In essence, to determine the rejection of \( H_0 \), it鈥檚 necessary to
    • Choose your significance level wisely, based on what's acceptable for your study.
    • Remember: The stricter the significance level, the stronger the evidence required.

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    Most popular questions from this chapter

    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 \(\mathrm{S} \& \mathrm{P}\) Stock Index of all major stocks, the mean \(\mathrm{P} / \mathrm{E}\) ratio is \(\mu=19.4 . \mathrm{A}\) 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\&CP Stock Index? Use \(\alpha=0.05\).

    Education influences attitude and lifestyle. Differences in education are a big factor in the "generation gap." Is the younger generation really better educated? Large surveys of people age 65 and older were taken in \(n_{1}=32\) U.S. cities. The sample mean for these cities showed that \(\bar{x}_{1}=15.2 \%\) of the older adults had attended college. Large surveys of young adults (age \(25-34\) ) were taken in \(n_{2}=35\) U.S. cities. The sample mean for these cities showed that \(\bar{x}_{2}=19.7 \%\) of the young adults had attended college. From previous studies, it is known that \(\sigma_{1}=7.2 \%\) and \(\sigma_{2}=5.2 \%\) (Reference: American Generations, S. Mitchell). Does this information indicate that the population mean percentage of young adults who attended college is higher? Use \(\alpha=0.05\).

    Again suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank (see Problem 5). You draw a random sample of \(n=228\) numbers from this file and \(r=92\) have a first nonzero digit of \(1 .\) Let \(p\) represent the population proportion of all numbers in the computer file that have a leading digit of 1 . i. Test the claim that \(p\) is more than \(0.301\). Use \(\alpha=0.01\). ii. If \(p\) is in fact larger than \(0.301\), it would seem there are too many numbers in the file with leading 1's. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees. 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?

    What is your favorite color? A large survey of countries, including the United States, China, Russia, France, Turkey, Kenya, and others. indicated that most people prefer the color blue. In fact, about \(24 \%\) of the population claim blue as their favorite color. (Reference: Study by \(J .\) Bunge and \(A\). Freeman-Gallant, Statistics Center, Cornell University.) Suppose a random sample of \(n=56\) college students were surveyed and \(r=12\) of them said that blue is their favorite color. Does this information imply that the color preference of all college students is different (either way) from that of the general population? Use \(\alpha=0.05\).

    Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha\) ? (e) State your conclusion in the context of the application. Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from this horse (in \(\mathrm{mg} / 100 \mathrm{ml}\) ). \(\begin{array}{llllllll}93 & 88 & 82 & 105 & 99 & 110 & 84 & 89\end{array}\) The sample mean is \(\bar{x} \approx 93.8\). Let \(x\) be a random variable representing glucose readings taken from Gentle Ben. We may assume that \(x\) has a normal distribution, and we know from past experience that \(\sigma=12.5 .\) The mean glucose level for horses should be \(\mu=85 \mathrm{mg} / 100 \mathrm{ml}\) (Reference: Merck Veterinary Manual). Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use \(\alpha=0.05\).
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