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Chapter 19: Problem 5

Which of the following statements are true? If false, explain briefly. a. Using an alpha level of \(0.05,\) a P-value of 0.04 results in rejecting the null hypothesis. b. The alpha level depends on the sample size. C. With an alpha level of \(0.01,\) a \(P\) -value of 0.10 results in rejecting the null hypothesis. d. Using an alpha level of \(0.05,\) a P-value of 0.06 means the null hypothesis is true.

Short Answer

Expert verified
a. True, b. False - alpha level is not dependent on sample size, c. False - P-value is greater than alpha level, d. False - high P-value does not confirm the null hypothesis.

Step by step solution

01

Evaluation of Statement a

The statement is true. According to the rule of hypothesis testing, if a P-value is less than or equal to alpha level of 0.05, null hypothesis will be rejected. Since P-value 0.04 is less than the alpha level, we reject the null hypothesis.
02

Evaluation of Statement b

This statement is false. The alpha level does not depend on the sample size, but rather it is a subjective threshold set by the researcher. Usually, common values are 0.05 or 0.01, but it does not depend on the size or characteristic of the sample used.
03

Evaluation of statement c

The statement is false. Given an alpha level of 0.01, a P-value of 0.10 is greater than the alpha level, therefore, we do not reject the null hypothesis.
04

Evaluation of Statement d

The statement is false. Using an alpha level of 0.05, a P-value of 0.06 does not mean the null hypothesis is true. It rather means we do not have enough solid empirical evidence to reject the null hypothesis. A high P-value does not imply that the null hypothesis is necessarily true, rather it might indicate that our sample has failed to provide enough statistical evidence against the null hypothesis.

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Key Concepts

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

Alpha Level
The alpha level, often denoted by the Greek letter \( \alpha \), represents the threshold for deciding whether the evidence against the null hypothesis is strong enough to reject it. This is typically set by the researcher before conducting a hypothesis test.
  • Common alpha level values are 0.05, 0.01, and sometimes 0.10.
  • It signifies the probability of making a Type I error, which is rejecting a true null hypothesis.
  • The alpha level does not change with the sample size; it remains a constant decision rule through the testing process.
Choosing an appropriate alpha level is crucial in deciding the stringency of a statistical test, reflecting the trade-off between being too lenient and overly strict.
P-value
A P-value measures the probability that an observed effect, or one more extreme, would occur, assuming the null hypothesis is true.
  • It is a continuous value, often compared directly to the predetermined alpha level.
  • If the P-value is less than or equal to the alpha level, you would reject the null hypothesis.
  • A lower P-value indicates stronger evidence against the null hypothesis.
It is important to remember that a P-value doesn鈥檛 tell us the probability of the null hypothesis itself being true or false. Instead, it provides a mechanism to weigh evidence given the assumption that the null hypothesis is true.
Null Hypothesis
The null hypothesis, denoted as \( H_0 \), is a basic assumption about a population from which a researcher starts. Often, it asserts that there is no effect or difference.
  • For example, it might state that two means are equal across different groups.
  • The null hypothesis is assumed to be true until evidence suggests otherwise.
  • It involves statistical tests to determine whether the data provides sufficient evidence to reject it.
Rejecting the null hypothesis implies that an alternative hypothesis, denoted \( H_a \), is more likely correct, leading to the conclusion of a significant effect or relationship.
Sample Size
Sample size refers to the number of observations or data points collected for a study. It plays a crucial role in the reliability and validity of hypothesis testing results.
  • Having a large enough sample size increases the power of a hypothesis test.
  • It helps in more precisely estimating population parameters.
  • Sample size affects the variability of the test and the generalizability of results.
While a larger sample size can lead to more reliable outcomes, it's also important to balance practical constraints, like time and cost, in designing the study. The alpha level remains a separate decision factor determined independently of the sample size.

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Most popular questions from this chapter

For each of the following situations, state whether a Type I, a Type II, or neither error has been made. a. A test of \(\mathrm{H}_{0}: \mu=25\) vs. \(\mathrm{H}_{\mathrm{A}}: \mu>25\) rejects the null hypothesis. Later it is discovered that \(\mu=24.9\). b. A test of \(\mathrm{H}_{0}: p=0.8\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.8\) fails to reject the null hypothesis. Later it is discovered that \(p=0.9\). c. A test of \(\mathrm{H}_{0}: p=0.5\) vs. \(\mathrm{H}_{\mathrm{A}}: p \neq 0.5\) rejects the null hypothesis. Later it is discovered that \(p=0.65\). d. A test of \(\mathrm{H}_{0}: p=0.7\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.7\) fails to reject the null hypothesis. Later it is discovered that \(p=0.6\).

Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person's income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person's score falls below the minimum cutoff will the bank reject the null and deny the loan. This system is reasonably reliable, but, of course, sometimes there are mistakes. a. When a person defaults on a loan, which type of error did the bank make? b. Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it? c. Suppose the bank decides to lower the cutoff score from 250 points to 200 . Is that analogous to choosing a higher or lower value of \(a\) for a hypothesis test? Explain. d. What impact does this change in the cutoff value have on the chance of each type of error?

A researcher developing scanners to search for hidden weapons at airports has concluded that a new device is significantly better than the current scanner. He made this decision based on a test using \(\alpha=0.05 .\) Would he have made the same decision at \(\alpha=0.10 ?\) How about \(\alpha=0.01 ?\) Explain.

Canine hip dysplasia is a degenerative disease that causes pain in many dogs. Sometimes advanced warning signs appear in puppies as young as 6 months. A veterinarian checked 42 puppies whose owners brought them to a vaccination clinic, and she found 5 with early hip dysplasia. She considers this group to be a random sample of all puppies. a. Explain why we cannot use this information to construct a confidence interval for the rate of occurrence of early hip dysplasia among all 6 -month- old puppies. b. Could you use a bootstrap hypothesis test? Why or why not?

An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About \(40 \%\) break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay from another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. a. Suppose the new, expensive clay really is no better than her usual clay. What's the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.) b. If she decides to switch to the new clay and it is no better, what kind of error did she commit? c. If the new clay really can reduce breakage to only \(20 \%,\) what's the probability that her test will not detect the improvement? d. How can she improve the power of her test? Offer at least two suggestions.
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