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Chapter 9: Problem 3

Explain how the tails of a test depend on the sign in the alternative hypothesis. Describe the signs in the null and alternative hypotheses for a two-tailed, a left-tailed, and a right-tailed test, respectively.

Short Answer

Expert verified
The 'tails' in a test refers to the ends of the distribution where the null hypothesis can be rejected, dictated by the statement of the alternative hypothesis. For a two-tailed test, the null hypothesis, H0, posits a parameter equals a value, and the alternative hypothesis (Ha) claims the parameter is not equal to this value. In a one-tailed (left-tailed) test, H0 says a parameter equals a value, and Ha presumes the parameter is less than this value. Rounding off, a one-tailed (right-tailed) test has H0 stating a parameter equals a value and Ha positing the parameter is greater than that value.

Step by step solution

01

Understand the concept of tails in a hypothesis test

The tails of a statistical test are the extreme ends of the distribution where, under the null hypothesis, we would not expect our test statistic to lie. These show where we can reject the null hypothesis. The number of tails depends on the statement of the alternative hypothesis. An alternative hypothesis that states the parameter is not equal to the value in the null hypothesis is two-tailed (it does not specify a direction). If the alternative hypothesis suggests the parameter is greater or less than the null hypothesis value, it is a one-tailed test (either right or left, respectively).
02

Describing null and alternative hypotheses for a two-tailed test

For a two-tailed test, the null hypothesis is often stated as the parameter being equal to a certain value, and the alternative hypothesis as the parameter not being equal to that value. The exact signs depend on the specific parameter under study, but generically we could write: H0: Parameter = Value, Ha: Parameter 鈮� Value.
03

Describing null and alternative hypotheses for a left-tailed test

For a left-tailed test, typically the null hypothesis states the parameter is equal to a stated value and the alternative hypothesis is that the parameter is less than that stated value. Generally, it looks like: H0: Parameter = Value, Ha: Parameter < Value.
04

Describing null and alternative hypotheses for a right-tailed test

For a right-tailed test, the null hypothesis typically posits the parameter being equal to a given value, and the alternative hypothesis suggests the parameter is greater than that value: H0: Parameter = Value, Ha: Parameter > Value.

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Key Concepts

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

Alternative Hypothesis
The alternative hypothesis is a key part of hypothesis testing. It represents a new claim we want to test against the null hypothesis, which is considered the default or status quo. In a hypothesis test, the alternative hypothesis suggests a specific statistical relationship that is different from the null hypothesis. This could mean the parameter is either not equal, less than, or greater than a specific value, depending on the type of test.
  • If the alternative hypothesis states "not equal," it indicates a two-tailed test.
  • "Greater than" suggests a right-tailed test.
  • "Less than" points to a left-tailed test.
The choice of alternative hypothesis directly influences the type of statistical test conducted and determines where in the distribution you'll be looking to reject the null hypothesis.
Null Hypothesis
The null hypothesis serves as the foundation of hypothesis testing. It is a statement of no effect or no difference and acts as a counterpoint to the alternative hypothesis. The null hypothesis is usually formulated to demonstrate that any observed effect is not significant and is due to chance. This hypothesis is generally expressed in an "equal to" format.
  • For a two-tailed test, it suggests the parameter is equal to a certain value.
  • In left-tailed and right-tailed tests, it assumes the parameter remains at or above the tested value without any deviation proposed by the alternative hypothesis.
This hypothesis is the assumption that hovers over your statistical test until enough evidence is found against it. Only with sufficient data do we reject the null hypothesis in favor of the alternative hypothesis.
Two-tailed Test
A two-tailed test is a common approach in hypothesis testing. It evaluates whether the parameter in question is significantly higher or lower than a certain value. It is most often used when researchers do not predict the direction of the deviation beforehand.
The two-tailed nature comes from the alternative hypothesis stating the parameter "is not equal to" the null hypothesis value. Thus, you check both ends of the distribution:
  • If a test statistic falls in the extreme parts of either tail, you have evidence to reject the null hypothesis.
  • It accommodates deviations in both directions, making it more conservative compared to one-tailed tests.
In practice, this means a larger confidence region, which helps in guarding against finding misleading significant results just because of random sampling.
Left-tailed Test
A left-tailed test is used when the alternative hypothesis indicates that the parameter of interest is less than a specified value.
This means that the null hypothesis posits the parameter as equal to a value, and the test looks for evidence to show the parameter is actually less.
  • The p-value calculation focuses on the lower tail of the probability distribution.
  • If the test statistic falls into this critical lower region, it suggests a significant difference from the null hypothesis.
Left-tailed tests are particularly beneficial in scenarios where a decrease in a parameter could indicate an important change, such as a decrease in production defects or a reduction in disease incidence.
Right-tailed Test
The right-tailed test is used when the alternative hypothesis suggests that a parameter is greater than the specified null hypothesis value.
The null hypothesis asserts the parameter as equal to the set value, whereas the alternative proposes it's more than this threshold. This leads to checking the upper end of the distribution:
  • This test places concern on the high side of the distribution curve.
  • A significant result is found if the test statistic lies on the far right tail.
Right-tailed tests are essential in situations where an increase in the parameter's value might be significant, such as growth in productivity measures or inflation rates above a policy's acceptable level.

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Most popular questions from this chapter

A business school claims that students who complete a 3 -month typing course can type, on average, at least 1200 words an hour. A random sample of 25 students who completed this course typed, on average, 1125 words an hour with a standard deviation of 85 words. Assume that the typing speeds for all students who complete this course have an approximately normal distribution. a. Suppose the probability of making a Type I error is selected to be zero. Can you conclude that the claim of the business school is true? Answer without performing the five steps of a test of hypothesis. b. Using the \(5 \%\) significance level, can you conclude that the claim of the business school is true? Use both approaches.

Shulman Steel Corporation makes bearings that are supplied to other companies. One of the machines makes bearings that are supposed to have a diamcter of 4 inches. The bearings that have a diameter of either more or less than 4 inches are considered defective and are discarded. When working properly, the machine does not produce more than \(7 \%\) of bearings that are defective. The quality control inspector selects a sample of 200 bearings each week and inspects them for the size of their diameters. Using the sample proportion, the quality control inspector tests the null hypothesis \(p \leq .07\) against the alternative hypothesis \(p \geq\) 07, where \(p\) is the proportion of bearings that are defective. He always uses a \(2 \%\) significance level. If the null hypothesis is rejected, the machine is stopped to make any necessary adjustments. One sample of 200 bearings taken recently contained 22 defective bearings. a. Using the \(2 \%\) significance level, will you conclude that the machine should be stopped to make necessary adjustments? b. Perform the test of part a using a \(1 \%\) significance level. Is your decision different from the one in part a?

As noted in U.S. Senate Resolution \(28,9.3 \%\) of Americans speak their native language and another language fluently (Source: www.actfl.org/i4a/pages/index.cfm?pageid=3782). Suppose that in a recent sample of 880 Americans, 69 speak their native language and another language fluently. Is there significant evidence at the \(10 \%\) significance level that the percentage of all Americans who speak their native language and another language fluently is different from \(9.3 \%\) ? Use both the \(p\) -value and the critical-value approaches.

The standard therapy used to treat a disorder cures \(60 \%\) of all patients in an average of 140 visits. A health care provider considers supporting a new therapy regime for the disorder if it is effective in reducing the number of visits while retaining the cure rate of the standard therapy. A study of 200 patients with the disorder who were treated by the new therapy regime reveals that 108 of them were cured in an average of 132 visits with a standard deviation of 38 visits. What decision should be made using a \(.01\) level of significance?

The print on the packages of 100-watt General Electric soft-white lightbulbs states that these lightbulbs have an average life of 750 hours. Assume that the standard deviation of the lengths of lives of these lightbulbs is 50 hours. A skeptical consumer does not think these lightbulbs last as long as the manufacturer claims, and she decides to test 64 randomly selected lightbulbs. She has set up the decision rule that if the average life of these 64 lightbulbs is less than 735 hours, then she will conclude that GE has printed too high an average length of life on the packages and will write them a letter to that effect. Approximately what significance level is the consumer using? Approximately what significance level is she using if she decides that GE has printed too high an average length of life on the packages if the average life of the 64 lightbulbs is less than 700 hours? Interpret the values you get.
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