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Chapter 8: Problem 23

(a) For the same data and null hypothesis, is the \(P\) -value of a one-tailed test (right or left) larger or smaller than that of a two-tailed test? Explain. (b) For the same data, null hypothesis, and level of significance, is it possible that a one-tailed test results in the conclusion to reject \(H_{0}\) while a twotailed test results in the conclusion to fail to reject \(H_{0}\) ? Explain. (c) For the same data, null hypothesis, and level of significance, if the conclusion is to reject \(H_{0}\) based on a two-tailed test, do you also reject \(H_{0}\) based on a one-tailed test? Explain. (d) If a report states that certain data were used to reject a given hypothesis, would it be a good idea to know what type of test (one-tailed or twotailed) was used? Explain.

Short Answer

Expert verified
(a) Smaller for one-tailed; (b) Yes, possible; (c) Yes, if rejected in two-tailed; (d) Yes, crucial to know.

Step by step solution

01

Understanding One-Tailed and Two-Tailed Tests

In a one-tailed test, the critical area of a distribution is one-sided, meaning the test is looking for evidence of an effect in only one direction. In contrast, a two-tailed test considers both directions, splitting the level of significance between two tails.
02

Comparing P-values

For the same data and null hypothesis, the P-value of a one-tailed test is typically smaller than that of a two-tailed test. This is because a one-tailed test only takes into consideration the extreme value in one direction, making it easier to reject the null hypothesis compared to a two-tailed test, which looks both ways.
03

Condition for Different Conclusions

A one-tailed test can result in the rejection of the null hypothesis while a two-tailed test does not, particularly if the observed data significantly deviates in the direction expected in the one-tailed test. The reduced significance level on each tail in the two-tailed test makes it harder to reach the critical region compared to a one-tailed test.
04

Rejection in Two-Tailed Test Implies Rejection in One-Tailed Test

If a two-tailed test leads to the rejection of the null hypothesis, then a one-tailed test, which focuses solely on the extreme in one direction, will also reject the null hypothesis assuming the data indicates deviation in that specific direction.
05

Importance of Knowing the Test Type

Knowing whether a test is one-tailed or two-tailed is crucial because it influences the hypothesis test's framework and interpretation. A one-tailed test is more powerful for detecting an effect in a specified direction but does not account for abnormalities in the opposite direction, whereas a two-tailed test could detect deviations in either direction.

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Key Concepts

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

One-tailed Test
In a hypothesis test, when we explore only one direction of the effect, we apply a one-tailed test. This type of test is designed to identify if there is a significant difference in a specific direction; for example, whether a mean is greater or less than a certain value.

The main advantage of using a one-tailed test is its power. Since it focuses only on one tail of the distribution, it allows us to detect an effect with a smaller critical value compared to a two-tailed test. This makes it easier to reject the null hypothesis if the effect happens in the expected direction.

However, if the effect occurs in the opposite direction, a one-tailed test will not reveal that, which might be a limitation. Therefore, it's crucial to determine the direction of the effect before selecting a one-tailed test, ensuring the test aligns accurately with the hypothesis being examined.
Two-tailed Test
When unsure about the direction of an effect, a two-tailed test is appropriate. It checks for the possibility of an effect in both directions by dividing the significance level between two tails of the bell curve.

A two-tailed test is more conservative than a one-tailed test because it requires stronger evidence to reject the null hypothesis. Since the level of significance is split across two tails, each tail has a smaller threshold.

This means you need more significant data to support rejecting the null hypothesis.

The two-tailed test is often used when researchers want to explore any deviation from the null hypothesis, regardless of direction. It provides a comprehensive analysis but might miss finer directional deviations that a one-tailed test would catch.
P-value
The P-value is a crucial element in hypothesis testing. It quantifies the probability of obtaining test results at least as extreme as the ones observed, assuming that the null hypothesis is true.

A P-value helps determine whether an observed effect is statistically significant. In a one-tailed test, because the focus is only on one direction, the P-value is calculated for that direction alone.

This generally results in a smaller P-value, making it easier to reach the statistical threshold needed to reject the null hypothesis.

In contrast, a two-tailed test considers extreme values in both directions by halving the significance level in each tail. Consequently, the P-value is usually larger compared to a one-tailed test P-value.
  • If the P-value is less than the chosen significance level, we reject the null hypothesis.
  • In hypothesis testing, knowing the P-value and the test direction is essential for interpreting whether an effect exists and its potential implications.

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Most popular questions from this chapter

Preference: Color 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 \(\alpha=0.05.\)

Plato's Dialogues: Prose Rhythm Symposium is part of a larger work referred to as Plato's Dialogues. Wishart and Leach (see source in Problem 15 ) found that about \(21.4 \%\) of five-syllable sequences in Symposium are of the type in which four are short and one is long. Suppose an antiquities store in Athens has a very old manuscript that the owner claims is part of Plato's Dialogues. A random sample of 493 five-syllable sequences from this manuscript showed that 136 were of the type four short and one long. Do the data indicate that the population proportion of this type of five-syllable sequence is higher than that found in Plato's Symposium? Use \(\alpha=0.01.\)

Wildlife: Wolves The following is based on information from The Wolf in the Southwest: The Making of an Endangered Species by David E. Brown (University of Arizona Press). Before \(1918,\) the proportion of female wolves in the general population of all southwestern wolves was about \(50 \% .\) However, after 1918, southwestern cattle ranchers began a widespread effort to destroy wolves. In a recent sample of 34 wolves, there were only 10 females. One theory is that male wolves tend to return sooner than females to their old territories where their predecessors were exterminated. Do these data indicate that the population proportion of female wolves is now less than \(50 \%\) in the region? Use \(\alpha=0.01.\)

Medical: Hypertension This problem is based on information taken from The Merck Manual (a reference manual used in most medical and nursing schools). Hypertension is defined as a blood pressure reading over \(140 \mathrm{mm}\) Hg systolic and/or over \(90 \mathrm{mm}\) Hg diastolic. Hypertension, if not corrected, can cause long-term health problems. In the college-age population \((18-24\) years), about \(9.2 \%\) have hypertension. Suppose that a blood donor program is taking place in a college dormitory this week (final exams week). Before each student gives blood, the nurse takes a blood pressure reading. Of 196 donors, it is found that 29 have hypertension. Do these data indicate that the population proportion of students with hypertension during final exams week is higher than \(9.2 \% ?\) Use a \(5 \%\) level of significance.

Consider a hypothesis test of difference of means for two independent populations \(x_{1}\) and \(x_{2}\) (a) What does the null hypothesis say about the relationship between the two population means? (b) If the sample test statistic has a \(z\) distribution, give the formula for \(z\) (c) If the sample test statistic has a \(t\) distribution, give the formula for \(t\).
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