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Chapter 8: Problem 1

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.

Short Answer

Expert verified
Null and alternative hypotheses are given based on whether the claim is one-tailed or two-tailed.

Step by step solution

01

Define Hypotheses for Part a

For this part, we have the historical average July temperature of \(74.5^{\circ} \mathrm{F}\). We want to determine if the current average is higher, which suggests a one-tailed test.**Null Hypothesis (\(H_0\))**: \( \mu = 74.5 \)**Alternative Hypothesis (\(H_a\))**: \( \mu > 74.5 \)
02

Define Hypotheses for Part b

For this part, the average weight of a female airline passenger with luggage was 145 pounds ten years ago. We want to determine if it's higher now, suggesting another one-tailed test.**Null Hypothesis (\(H_0\))**: \( \mu = 145 \)**Alternative Hypothesis (\(H_a\))**: \( \mu > 145 \)
03

Define Hypotheses for Part c

For this part, the average stipend for doctoral students is \(\$14,756\). The department chairman believes the national average is higher, which again implies a one-tailed test.**Null Hypothesis (\(H_0\))**: \( \mu = 14756 \)**Alternative Hypothesis (\(H_a\))**: \( \mu > 14756 \)
04

Define Hypotheses for Part d

For this part, the average room rate in hotels is \(\$82.53\). The belief is that the average in a particular resort area is different, implying a two-tailed test.**Null Hypothesis (\(H_0\))**: \( \mu = 82.53 \)**Alternative Hypothesis (\(H_a\))**: \( \mu eq 82.53 \)
05

Define Hypotheses for Part e

For this part, the average farm size was 69.4 acres. The assertion is that it is less today, suggesting a one-tailed test.**Null Hypothesis (\(H_0\))**: \( \mu = 69.4 \)**Alternative Hypothesis (\(H_a\))**: \( \mu < 69.4 \)

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Key Concepts

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

Null Hypothesis
In any hypothesis test, the null hypothesis ( H_0 ) serves as the statement we initially assume to be true. It is a 'no effect' or 'status quo' proposition that we attempt to find evidence against. The null hypothesis typically states that there is no change or difference from the known or historical value. For example, when considering a scenario where we check if the current average temperature deviates from the historical 74.5掳F, the null hypothesis would assert that the average temperature remains at 74.5掳F. This serves as the benchmark against which alternative claims are tested.
  • Stable Assertion: Example scenarios reiterate past data as still valid.
  • Test of Change: It helps to validate assumptions against collected data.
Alternative Hypothesis
The alternative hypothesis ( H_a ) directly contrasts with the null hypothesis. It reflects the statement we want to test and is usually the claim that needs backing up with statistical evidence. The alternative hypothesis introduces a change or a new observation that challenges the status quo expressed in the null hypothesis. In a situation where the company believes the average room rate is different from $82.53, the alternative hypothesis would state that the average rate is not the same as this known value.
  • Expresses Difference or Change: Proposes specifics such as increase, decrease, or not equal to.
  • Requires Evidence: Needs data support to reject the null hypothesis.
One-tailed Test
A one-tailed test evaluates the direction of an effect, checking if a parameter is either greater than or less than a specific value. It moves in one direction. This test is used when we have a specific hypothesis about the direction of an effect. For instance, if we hypothesize that the average weight of passengers is greater than it was ten years ago, a one-tailed test would assess evidence supporting this specific direction of change.
  • Directional Hypothesis: Specifically tests greater-than or less-than conditions.
  • Efficiency: More powerful for detecting an effect in a specified direction.
Common in situations where a precise direction of change is expected, one-tailed tests can offer clearer insights when only one outcome is of interest.
Two-tailed Test
A two-tailed test assesses whether there is any change or difference, without focusing on the supposed direction of that change. It's employed when we want to determine if a value is simply different from a benchmark, whether higher or lower. This type of test is suitable when there are no assumptions about which direction the test will take. For example, if there's doubt about whether the average room rate in a resort area is different from $82.53, a two-tailed test would evaluate evidence for change in both directions.
  • Non-directional Test: Checks for any departure from the null, either high or low.
  • Comprehensive: Effectively captures any form of deviation from expected value.
Two-tailed tests provide a broader check against the null hypothesis, often employed when no exact increase or decrease is assumed.

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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 \(H_{0} \mu=\mu_{0}\) 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.

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).

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: \(\mu \neq 78.8 @ \alpha=0.10\).

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. \(H a: \mu>3.8\) @ \(\alpha=0.001\)
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