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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
a) Valid b) Valid c) Invalid d) Valid

Step by step solution

01

Evaluate first set

For the first set of hypothesis, we have \(H_{0}: \rho=0\) and \(H_{a}: \rho<0\). It is acceptable because if the null hypothesis is rejected, the alternative is automatically considered to be true. Thus, it's a valid set of hypotheses.
02

Evaluate second set

For the second set \(H_{0}: \hat{p}=0.3\) and \(H_{a}: \hat{p} \neq 0.3\), we can also observe this sort of opposition between the null hypothesis and alternative hypothesis, thus they are valid.
03

Evaluate third set

For hypothesis pair \(H_{0}: \mu_{1} \neq \mu_{2}\) and \(H_{a}: \mu_{1}=\mu_{2}\), the null hypothesis assumes inequality, while the alternative assumes equality. This is the opposite of the usual setup, hence it is not a valid set. Null hypothesis should always contain equal sign.
04

Evaluate fourth set

For the fourth and final set, \(H_{0}: p=25\) and \(H_{a}: p \neq 25\), there is a clear opposition between the null and alternative hypothesis where the null hypothesis states that the parameter is equal to a specific value, and the alternative hypothesis states that it does not equal that value. Hence, this represents a valid set of hypotheses.

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Key Concepts

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

Understanding the Null Hypothesis
The null hypothesis, commonly denoted as \(H_0\), is a statement used in statistics that represents a default position or assumption. In simpler terms, it is what you presume to be true unless you have strong evidence to the contrary. For example, if a new medicine is claimed to have an effect, the null hypothesis might state that it actually has no effect.
  • The null hypothesis provides a baseline to compare against.
  • It is often expressed as an equivalence or no-effect statement, such as \(\rho = 0\) or \(p = 25\).
The critical part about the null hypothesis is that it should include an equal sign. This is important because when conducting a statistical test, you actually test the strength of evidence against this null hypothesis. In the exercise given, the third set \((H_0: \mu_1 eq \mu_2)\) is considered invalid because the null hypothesis should not state inequality.
Exploring the Alternative Hypothesis
The alternative hypothesis, denoted as \(H_a\) or \(H_1\), is a statement used in statistics that contradicts the null hypothesis. It's what researchers generally want to prove. For instance, in drug testing, while the null hypothesis might state that a drug has no effect, the alternative hypothesis suggests that there is an effect.
  • The alternative presents a scenario where there is a significant effect or difference.
  • It typically includes inequality, such as \(\rho < 0\) or \(\hat{p} eq 0.3\).
When statisticians conduct experiments or analyses, their aim is to collect data sufficient to reject the null hypothesis, thus providing support for the alternative. Each valid hypothesis pair provided in the initial exercise has this opposing nature between the null and alternative hypotheses that complements the correctness of the relationships outlined.
Examining the Validity of Hypotheses
Determining whether a set of hypotheses is valid is crucial in statistical testing. Valid hypotheses create a proper framework for testing an experiment or observation. They must adhere to a specific format where:
  • The null hypothesis serves as a statement of no effect or no difference.
  • The alternative hypothesis must oppose the null.
Moreover, for a set of hypotheses to be termed valid:
- The null must include an equality sign.- The statements should not be contradictory to the standard setup.In the examples given, the incorrect setup in \(H_0: \mu_1 eq \mu_2\) versus \(H_a: \mu_1 = \mu_2\) does not follow these conventions as it incorrectly aligns the assumptions. Ensuring hypotheses are arranged correctly allows for meaningful and interpretable statistical results.

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Most popular questions from this chapter

Exercise 2.19 on page 58 introduces a study examining whether giving antibiotics in infancy increases the likelihood that the child will be overweight. Prescription records were examined to determine whether or not antibiotics were prescribed during the first year of a child's life, and each child was classified as overweight or not at age 12. (Exercise 2.19 looked at the results for age 9.) The researchers compared the proportion overweight in each group. The study concludes that: "Infants receiving antibiotics in the first year of life were more likely to be overweight later in childhood compared with those who were unexposed \((32.4 \%\) versus \(18.2 \%\) at age 12 \(P=0.002) "\) (a) What is the explanatory variable? What is the response variable? Classify each as categorical or quantitative. (b) Is this an experiment or an observational study? (c) State the null and alternative hypotheses and define the parameters. (d) Give notation and the value of the relevant sample statistic. (e) Use the p-value to give the formal conclusion of the test (Reject \(H_{0}\) or Do not reject \(H_{0}\) ) and to give an indication of the strength of evidence for the result. (f) Can we conclude that whether or not children receive antibiotics in infancy causes the difference in proportion classified as overweight?

Eating Breakfast Cereal and Conceiving Boys Newscientist.com ran the headline "Breakfast Cereals Boost Chances of Conceiving Boys," based on an article which found that women who eat breakfast cereal before becoming pregnant are significantly more likely to conceive boys. \({ }^{42}\) The study used a significance level of \(\alpha=0.01\). The researchers kept track of 133 foods and, for each food, tested whether there was a difference in the proportion conceiving boys between women who ate the food and women who didn't. Of all the foods, only breakfast cereal showed a significant difference. (a) If none of the 133 foods actually have an effect on the gender of a conceived child, how many (if any) of the individual tests would you expect to show a significant result just by random chance? Explain. (Hint: Pay attention to the significance level.) (b) Do you think the researchers made a Type I error? Why or why not? (c) Even if you could somehow ascertain that the researchers did not make a Type I error, that is, women who eat breakfast cereals are actually more likely to give birth to boys, should you believe the headline "Breakfast Cereals Boost Chances of Conceiving Boys"? Why or why not?

Income East and West of the Mississippi For a random sample of households in the US, we record annual household income, whether the location is east or west of the Mississippi River, and number of children. We are interested in determining whether there is a difference in average household income between those east of the Mississippi and those west of the Mississippi. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) What statistic(s) from the sample would we use to estimate the difference?

Influencing Voters When getting voters to support a candidate in an election, is there a difference between a recorded phone call from the candidate or a flyer about the candidate sent through the mail? A sample of 500 voters is randomly divided into two groups of 250 each, with one group getting the phone call and one group getting the flyer. The voters are then contacted to see if they plan to vote for the candidate in question. We wish to see if there is evidence that the proportions of support are different between the two methods of campaigning. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Possible sample results are shown in Table 4.3 . Compute the two sample proportions: \(\hat{p}_{c},\) the proportion of voters getting the phone call who say they will vote for the candidate, and \(\hat{p}_{f},\) the proportion of voters getting the flyer who say they will vote for the candidate. Is there a difference in the sample proportions? (c) A different set of possible sample results are shown in Table 4.4. Compute the same two sample proportions for this table. (d) Which of the two samples seems to offer stronger evidence of a difference in effectiveness between the two campaign methods? Explain your reasoning. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Will Vote } \\ \text { Sample A } \end{array} & \text { for Candidate } & \begin{array}{l} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 152 & 98 \\ \text { Flyer } & 145 & 105 \\ \hline \end{array} $$ $$ \begin{array}{lcc} \text { Sample B } & \begin{array}{c} \text { Will Vote } \\ \text { for Candidate } \end{array} & \begin{array}{c} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 188 & 62 \\ \text { Flyer } & 120 & 130 \\ \hline \end{array} $$

Significant and Insignificant Results (a) If we are conducting a statistical test and determine that our sample shows significant results, there are two possible realities: We are right in our conclusion or we are wrong. In each case, describe the situation in terms of hypotheses and/or errors. (b) If we are conducting a statistical test and determine that our sample shows insignificant results, there are two possible realities: We are right in our conclusion or we are wrong. In each case, describe the situation in terms of hypotheses and/or errors. (c) Explain why we generally won't ever know which of the realities (in either case) is correct.
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