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Chapter 4: Problem 15

Determine whether the sets of hypotheses given are valid hypotheses. State whether each set of hypotheses is valid for a statistical test. If not valid, explain why not. (a) \(H_{0}: \rho=0 \quad\) vs \(\quad H_{a}: \rho<0\) (b) \(H_{0}: \hat{p}=0.3 \quad\) vs \(\quad H_{a}: \hat{p} \neq 0.3\) (c) \(H_{0}: \mu_{1} \neq \mu_{2} \quad\) vs \(\quad H_{a}: \mu_{1}=\mu_{2}\) (d) \(H_{0}: p=25 \quad\) vs \(\quad H_{a}: p \neq 25\)

Short Answer

Expert verified
The hypotheses sets (a), (b), and (d) are valid. Hypotheses set (c) is not valid because the null hypothesis isn't asserting a lack of effect.

Step by step solution

01

Evaluate Hypothesis (a)

For the hypothesis (a) \(H_{0}: \rho=0 \) vs \(H_{a}: \rho<0 \). This is a valid set of hypotheses. The null hypothesis assumes no effect (that is, no correlation), while the alternative hypothesis assumes that there is a negative correlation.
02

Evaluate Hypothesis (b)

For the hypothesis (b) \(H_{0}: \hat{p}=0.3 \) vs \(H_{a}: \hat{p} \neq 0.3 \). This is a valid set of hypotheses. The null hypothesis assumes the proportion is equal to 0.3, while the alternative hypothesis is that the proportion isn't equal to 0.3.
03

Evaluate Hypothesis (c)

The hypothesis (c) \(H_{0}: \mu_{1} \neq \mu_{2} \) vs \(H_{a}: \mu_{1}=\mu_{2} \) isn't valid. This is because the null hypothesis isn't asserting a lack of effect or difference, instead, it's asserting a difference exists. This goes against the standard practice of hypothesis testing, where the null hypothesis is about the lack of effect or difference.
04

Evaluate Hypothesis (d)

For the hypothesis (d) \(H_{0}: p=25 \) vs \(H_{a}: p \neq 25 \). This is a valid set of hypotheses. The null hypothesis asserts that some parameter p is equal to 25. The alternative asserts that p isn't equal to 25, covering all other possible outcomes. However, keep in mind that p must represent a parameter that could reasonably have a value of 25 in the context of the experiment; otherwise, these hypotheses wouldn't make sense.

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