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Chapter 10: Problem 24

State the null hypothesis, \(H_{o},\) and the alternative hypothesis, \(H_{a},\) that would be used to test these claims: a. The mean of the differences between the post-test and the pre-test scores is greater than \(15 .\) b. The mean weight gain, after the change in diet for the laboratory animals, is at least 10 oz. c. The mean weight loss experienced by people on a new diet plan was no less than 12 lb. d. The mean difference in the home reassessments from the two town assessors was no more than \(\$ 200\)

Short Answer

Expert verified
a. Null Hypothesis, \(H_{0}: \mu_{d} = 15\), Alternative Hypothesis, \(H_{a}: \mu_{d} > 15\). b. Null Hypothesis, \(H_{0}: \mu = 10\), Alternative Hypothesis, \(H_{a}: \mu > 10\). c. Null Hypothesis, \(H_{0}: \mu = 12\), Alternative Hypothesis, \(H_{a}: \mu \geq 12\). d. Null Hypothesis, \(H_{0}: \mu_{d} = 200\), Alternative Hypothesis, \(H_{a}: \mu_{d} \leq 200\).

Step by step solution

01

Formulate hypotheses for claim a

For claim a, where it's stated that the mean difference between the post-test and pre-test scores is greater than 15, the null and alternative hypotheses can be formulated as follows: - Null Hypothesis, \(H_{0}\): \(\mu_{d} = 15\), where \(\mu_{d}\) is the mean of differences. - Alternative Hypothesis, \(H_{a}\): \(\mu_{d} > 15\). This means we are testing if the mean of the differences is greater than 15.
02

Formulate hypotheses for claim b

For claim b, where a change in diet results in a mean weight gain of at least 10 oz, the null and alternative hypotheses are as follows: - Null Hypothesis, \(H_{0}\): \(\mu = 10\), \(\mu\) being the mean weight gain. - Alternative Hypothesis, \(H_{a}\): \(\mu > 10\). Here, we're checking if the mean weight gain exceeds 10 oz.
03

Formulate hypotheses for claim c

In claim c, it's given that the mean weight loss is no less than 12 lb. The null and alternative hypotheses are: - Null Hypothesis, \(H_{0}\): \(\mu = 12\), where \(\mu\) is the mean weight loss. - Alternative Hypothesis, \(H_{a}\): \(\mu \geq 12\). The test here is if the weight loss equals or surpasses 12lb.
04

Formulate hypotheses for claim d

Finally for claim d, where the mean difference in the home reassessments between the two assessors is no more than $200, the null and alternative hypotheses will be: - Null Hypothesis, \(H_{0}\): \(\mu_{d} = 200\), \(\mu_{d}\) being the mean difference. - Alternative Hypothesis, \(H_{a}\): \(\mu_{d} \leq 200\). Here, the hypothesis tested is whether the mean difference does not exceed $200.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis, often denoted as \(H_0\), serves as the baseline statement that there is no effect or no difference. It is the assumption made about a population parameter before any evidence is gathered from a sample.The null hypothesis is tested to determine its validity and is usually set up as a statement of equality. For instance:
  • For a claim that the mean difference is greater than 15, the null hypothesis could be \(H_0: \mu_d = 15\).
  • If a laboratory experiment suggests weight gain of at least 10 oz., the null hypothesis might state \(H_0: \mu = 10\).
Testing the null hypothesis involves collecting data and analyzing whether the observed evidence is strong enough to reject this assumption. To move proofs against the null hypothesis, data has to provide substantial evidence that favors an alternative explanation. If the evidence is not strong enough, the null hypothesis is not rejected, implying no significant changes were detected.
Alternative Hypothesis
The alternative hypothesis, indicated as \(H_a\), proposes what we suspect might be true instead of the null hypothesis. It is a statement indicating the presence of an effect or a difference that the researcher aims to test. Unlike the null hypothesis, it often implies an inequality.Here is how the alternative hypothesis can be structured:
  • For a pre-test and post-test score where the belief is the mean difference is greater than 15, the alternative hypothesis could be \(H_a: \mu_d > 15\).
  • In the case of weight gain from a diet, where it's expected to be more than 10 oz., \(H_a: \mu > 10\) can be the alternative hypothesis.
The purpose of testing the alternative hypothesis is to determine if there is enough statistical evidence to support the claim. The alternative hypothesis stands as the competitor to the null hypothesis, and successful evidence against \(H_0\) leads to the acceptance of \(H_a\). This acceptance suggests the observed data align with the claim made by the alternative hypothesis.
Mean Difference
The mean difference refers to the average difference between two sets of observations. In the context of hypothesis testing, it plays a significant role in understanding relationships or changes between samples.The mean difference is often used:
  • To compare scores before and after a particular intervention, such as a teaching method or treatment in clinical trials.
  • To assess differences in measurements between two groups, such as weight changes or score differences.
Calculating this involves subtracting one set of scores from another, then finding the average of these differences. In practice, hypotheses about the mean difference might relate to claims where the interest is in whether one metric significantly surpasses or undercuts a specific benchmark or another metric. For instance, if a researcher expects the mean difference between two assessments not to exceed a certain amount, then this expectation guides the hypothesis formulation, such as \(H_a: \mu_d \leq 200\) against a null set at \(H_0: \mu_d = 200\).Understanding and testing mean differences help in assessing claims, ensuring the reliability of experimental or observational insights provided by the data.

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Most popular questions from this chapter

At a large university, a mathematics placement exam is administered to all students. Samples of 36 male and 30 female students are randomly selected from this year's student body and the following scores recorded: $$\begin{array}{lllllllllll}\hline \text { Male } & 72 & 68 & 75 & 82 & 81 & 60 & 75 & 85 & 80 & 70 \\\& 71 & 84 & 68 & 85 & 82 & 80 & 54 & 81 & 86 & 79 \\\& 99 & 90 & 68 & 82 & 60 & 63 & 67 & 72 & 77 & 51 \\\& 61 & 71 & 81 & 74 & 79 & 76 & & & & \\\\\hline \text { Female } & 81 & 76 & 94 & 89 & 83 & 78 & 85 & 91 & 83 & 83 \\\& 84 & 80 & 84 & 88 & 77 & 74 & 63 & 69 & 80 & 82 \\\& 89 & 69 & 74 & 97 & 73 & 79 & 55 & 76 & 78 & 81 \\ \hline\end{array}$$ a. Describe each set of data with a histogram (use the same class intervals on both histograms), the mean, and the standard deviation. b. Construct \(95 \%\) confidence interval for the mean score for all male students. Do the same for all female students. c. Do the results found in part b show that the mean scores for males and females could be the same? Justify your answer. Be careful! d. Construct the \(95 \%\) confidence interval for the difference between the mean scores for male and female students. e. Do the results found in part d show that the mean scores for male and female students could be the same? Explain. f. Explain why the results in part b cannot be used to draw conclusions about the difference between the two means.

The proportions of defective parts produced by two machines were compared, and the following data were collected: Machine \(1: n=150 ;\) number of defective parts \(=12\) Machine \(2: n=150:\) number of defective parts \(=6\) Determine a \(90 \%\) confidence interval for \(p_{1}-p_{2}\).

Many cheeses are produced in the shape of a wheel, and due to manufacturing inconsistencies, the amount of cheese, measured by weight, varies from wheel to wheel. Heidi Cembert wishes to determine if there is a significant difference, at the \(10 \%\) level, between the weight per wheel of Gouda and Brie cheese. She randomly samples 16 wheels of Gouda and finds the mean to be 1.2 pounds with a standard deviation of 0.32 pound and then samples 14 wheels of Brie and finds the mean to be 1.05 pounds with a standard deviation of 0.25 pound. At the 0.05 level of significance, is there sufficient evidence to support Heidi's contention that there is a difference in the mean weights of the two types of cheese?

In trying to estimate the amount of growth that took place in the trees recently planted by the County Parks Commission, 36 trees were randomly selected from the 4000 planted. The heights of these trees were measured and recorded. One year later, another set of 42 trees was randomly selected and measured. Do the two sets of data (36 heights, 42 heights) represent dependent or independent samples? Explain.

Ten recently diagnosed diabetics were tested to determine whether an educational program would be effective in increasing their knowledge of diabetes. They were given a test, before and after the educational program, concerning self-care aspects of diabetes. The scores on the test were as follows: $$\begin{array}{lcccccccccc}\text { Patient } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Before } & 75 & 62 & 67 & 70 & 55 & 59 & 60 & 64 & 72 & 59 \\\\\text { Afler } & 77 & 65 & 68 & 72 & 62 & 61 & 60 & 67 & 75 & 68 \\\\\hline\end{array}$$The following MINITAB output may be used to determine whether the scores improved as a result of the program. Verify the values shown on the output [mean difference(MEAN), standard deviation of the difference (STDEV), standard error of the difference (SE MEAN), \(t \star\) (T-Value), and \(p\) -value] by calculating the values yourself.
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