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Chapter 8: Problem 2

Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two- tailed. a. \(\quad H 0: \mu=141\) VS. \(H a: \mu<141\) \(@ \alpha=0.20 .\) b. \(\quad H 0: \mu=-54\) VS. \(H a: \mu<-54\) @ \(\alpha=0.05 .\) C. \(\quad H 0: \mu=98.6\) VS. \(H a: \mu \neq 98.6\) \(@ \alpha=0.05 .\) d. \(\quad H 0: \mu=3.8\) VS. \(H a: \mu>3.8\) @ \(\alpha=0.001\)

Short Answer

Expert verified
(a) Left-tailed: \( z < -0.84 \); (b) Left-tailed: \( z < -1.645 \); (c) Two-tailed: \( z < -1.96 \) or \( z > 1.96 \); (d) Right-tailed: \( z > 3.09 \).

Step by step solution

01

Identify the Test Type for Hypothesis a

For the hypothesis test \( H_0: \mu = 141 \) vs. \( H_a: \mu < 141 \), this is a left-tailed test because the alternative hypothesis is testing if the population mean \( \mu \) is less than the hypothesized mean.
02

Determine the Rejection Region for Hypothesis a

With \( \alpha = 0.20 \) for a left-tailed test, the rejection region is to the left of the critical value found using a standard normal distribution table or calculator. The critical value for a left-tailed test with \( \alpha = 0.20 \) is approximately \( z = -0.84 \). Hence, the rejection region is \( z < -0.84 \).
03

Identify the Test Type for Hypothesis b

For the hypothesis test \( H_0: \mu = -54 \) vs. \( H_a: \mu < -54 \), this is a left-tailed test similar to part a since the alternative hypothesis suggests the mean is less than the hypothesized value.
04

Determine the Rejection Region for Hypothesis b

With \( \alpha = 0.05 \) for a left-tailed test, the critical value is found from standard normal distribution tables or calculators. The critical value is approximately \( z = -1.645 \). Thus, the rejection region is \( z < -1.645 \).
05

Identify the Test Type for Hypothesis c

For \( H_0: \mu = 98.6 \) vs. \( H_a: \mu eq 98.6 \), this is a two-tailed test because the alternative hypothesis tests if the population mean is different from the hypothesized mean.
06

Determine the Rejection Region for Hypothesis c

With \( \alpha = 0.05 \) in a two-tailed test, you split the significance level between the two tails, giving \( \alpha/2 = 0.025 \) for each tail. The critical values are approximately \( z = -1.96 \) and \( z = 1.96 \). Thus, the rejection regions are \( z < -1.96 \) and \( z > 1.96 \).
07

Identify the Test Type for Hypothesis d

For the hypothesis \( H_0: \mu = 3.8 \) vs. \( H_a: \mu > 3.8 \), this is a right-tailed test because the alternative hypothesis states that the mean is greater than the hypothesized value.
08

Determine the Rejection Region for Hypothesis d

With \( \alpha = 0.001 \) for a right-tailed test, find the critical value using standard normal distribution information. The critical value is approximately \( z = 3.09 \). So, the rejection region is \( z > 3.09 \).

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Key Concepts

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

Rejection Region
In hypothesis testing, the rejection region is a critical part of deciding whether to reject the null hypothesis. This region is determined based on the chosen significance level (\( \alpha \)) and the type of test being conducted. The rejection region represents the set of values for which the null hypothesis is not considered valid, whether it includes low or high extremity values, depending on the tail of the test.
  • Left-Tailed Test: In a left-tailed test, the rejection region is located in the lower tail of the normal distribution curve. For example, with a significance level of \( \alpha = 0.05 \), the rejection region might be at values like \( z < -1.645 \).
  • Right-Tailed Test: This type of test places the rejection region in the higher tail. For instance, with \( \alpha = 0.001 \), a rejection region could start at \( z > 3.09 \).
  • Two-Tailed Test: Here, the rejection regions exist in both tails of the distribution, such as \( z < -1.96 \) and \( z > 1.96 \) for \( \alpha = 0.05 \).
Breaking these regions down helps illuminate where statistically significant results might occur.
Normal Distribution
The normal distribution is fundamental in hypothesis testing because many test statistics assume this distribution. Also known as the bell curve, it is symmetric around the mean, where mean, median, and mode are equal. Its standard form is particularly useful because it allows statisticians to calculate probabilities and critical values needed for determining rejection regions.Several properties make the normal distribution particularly suitable for statistical analysis:
  • Symmetry: It ensures that results proportionally represent both lower and upper sides of the mean.
  • Standardization: Through conversion to standard normal variates \( (Z-scores) \), comparisons across different datasets become feasible.
  • Universality: Due to the Central Limit Theorem, the means of reasonably sized samples will approximate a normal distribution, making it a critical assumption in hypothesis testing.
The ability to utilize areas under this curve, through standard normal tables or calculators, facilitates determining whether test statistics fall in a rejection region.
Critical Value
The critical value is a key factor in hypothesis testing. It defines the boundary or threshold that separates the rejection region from the non-rejection region in the normal distribution. For any given significance level \( \alpha \), the critical value can be identified using a standard normal distribution table or calculator.Here鈥檚 how critical values work in different tests:
  • Determining Critical Value for Left-Tailed Tests: For a left-tailed test with \( \alpha = 0.05 \), the critical value is typically found to be around \( z = -1.645 \). Any test statistic falling below this value falls into the rejection region.
  • Determining Critical Value for Right-Tailed Tests: In right-tailed tests, such as one with \( \alpha = 0.001 \), a critical value might be \( z = 3.09 \), indicating greater values fall into the rejection region.
  • Determining Critical Values for Two-Tailed Tests: With \( \alpha = 0.05 \), critical values could be \( z = -1.96 \) and \( z = 1.96 \), indicating rejections in both donward and upward directions.
Correctly determining the critical value is vital for accurate hypothesis testing results.
Significance Level
The significance level, denoted as \( \alpha \), is an essential component in hypothesis testing. It indicates the probability of rejecting the null hypothesis when it is, in fact, true. Essentially, it determines how strict the test is regarding what is considered significant evidence against the null hypothesis.Common significance levels include 0.05, 0.01, and 0.001. Here鈥檚 their impact on tests:
  • Higher Significance Levels: For instance, \( \alpha = 0.20 \), as in some left-tailed tests, signals lower strictness, allowing a greater chance to reject the null hypothesis.
  • Moderate Significance Levels: The frequently used \( \alpha = 0.05 \) balances the risk of type I errors with sufficient strictness.
  • Low Significance Levels: A value like \( \alpha = 0.001 \) used in right-tailed tests, increases the stringency, lessening the probability of mistakenly rejecting the null hypothesis.
When selecting a significance level, it's crucial to weigh the consequences of errors, customizing it according to the context of the hypothesis test.

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Most popular questions from this chapter

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number \(\mu_{0}\) and write \(H_{0} \cdot \mu=\mu_{0}\) and the appropriate analogous expression for \(H_{a}\).) a. The average July temperature in a region historically has been \(74.5^{\circ} \mathrm{F}\). Perhaps it is higher now. b. The average weight of a female airline passenger with luggage was 145 pounds ten years ago. The FAA believes it to be higher now. c. The average stipend for doctoral students in a particular discipline at a state university is \(\$ 14,756\). The department chairman believes that the national average is higher. d. The average room rate in hotels in a certain region is \(\$ 82.53 .\) A travel agent believes that the average in a particular resort area is different. e. The average farm size in a predominately rural state was 69.4 acres. The secretary of agriculture of that state asserts that it is less today.

Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two- tailed. a. \(\quad H 0: \mu=141\) VS. Ha: \(\mu<141\) \(@ \alpha=0.20\). b. \(\quad H 0: \mu=-54\) vs. Ha: \(\mu<-54 @ \alpha=0.05\). C. \(\quad H 0: \mu=98.6\) VS. \(H a: \mu \neq 98.6 @ \alpha=0.05 .\) d. \(\quad H 0: \mu=3.8\) VS. Ha: \(\mu>3.8 @ \alpha=0.001\).

Find the rejection region (for the standardized test statistic) for each hypothesis test. a. \(\quad H 0: \mu=27\) VS. \(H a ; \mu<27\) \(@ \alpha=0.05\). b. \(\quad H 0: \mu=52\) vs. \(H a: \mu \neq 52\) \(@ \alpha=0.05 .\) c. \(\quad H 0: \mu=-105\) VS. Ha: \(\mu>-105\) \(@ \alpha=0.10\). d. \(\quad H 0: \mu=78.8\) VS. Ha: \(\mu \neq 78.8 @ \alpha=0.10\).

The mean yield for hard red winter wheat in a certain state is 44.8 bu/acre. In a pilot program a modified growing scheme was introduced on 35 independent plots. The result was a sample mean yield of 45.4 bu/acre with sample standard deviation 1.6 bu/acre, an apparent increase in yield. a. Test at the \(5 \%\) level of significance whether the mean yield under the new scheme is greater than 44.8 bu/acre, using the critical value approach. b. Compute the observed significance of the test. c. Perform the test at the \(5 \%\) level of significance using the \(p\) -value approach. You need not repeat the first three steps, already done in part (a).

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number \(\mu_{0}\) and write \(H_{0} \mu=\mu_{0}\) and the appropriate analogous expression for \(H_{a}\).) a. The average time workers spent commuting to work in Verona five years ago was 38.2 minutes. The Verona Chamber of Commerce asserts that the average is less now. b. The mean salary for all men in a certain profession is \(\$ 58,291\). A special interest group thinks that the mean salary for women in the same profession is different. c. The accepted figure for the caffeine content of an 8 -ounce cup of coffee is \(133 \mathrm{mg}\). A dietitian believes that the average for coffee served in a local restaurants is higher. d. The average yield per acre for all types of corn in a recent year was 161.9 bushels. An economist believes that the average yield per acre is different this year. e. An industry association asserts that the average age of all self-described fly fishermen is 42.8 years. A sociologist suspects that it is higher.
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