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Chapter 4: Problem 125

We are conducting many hypothesis tests to test a claim. In every case, assume that the null hypothesis is true. Approximately how many of the tests will incorrectly find significance? 300 tests using a significance level of \(1 \%\).

Short Answer

Expert verified
Approximately 3 tests will incorrectly find significance.

Step by step solution

01

Understand the Data

The data given is the total number of tests which is 300 and the significance level which is \(1\% = 0.01\). The null hypothesis is assumed to be true.
02

Calculate the Expected False Positives

To find out how many of the tests will incorrectly find significance (Type I error), multiply the total number of tests by the significance level. Mathematically, this can be written as: \[ \text{Number of Type I errors} = \text{Total number of tests} \times \text{Significance level} \] Substitute the given values into the equation: \[ \text{Number of Type I errors} = 300 \times 0.01 \]
03

Compute the Result

After performing the multiplication, get the approximate number of tests that will incorrectly find significance. Therefore, the solution to the problem is the result of this computation.

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Key Concepts

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

Hypothesis Testing
At the heart of hypothesis testing lies the null hypothesis, symbolized as ull hypothesisNull hypothesis hypothesis.
False Positives in Statistics
False positives throw a wrench into the works of statistical analysis. They're akin to a fire alarm going off without a trace of fire鈥攁 'Type I error.' In statistics, a false positive is concluding that an effect or difference exists when in fact, it doesn't, assuming that the null hypothesis is true.

The trouble with false positives is their potential to mislead. In medical testing, this could mean diagnosing a healthy patient with a condition they don't have. In legal terms, it's the innocent being wrongfully convicted. False positives not only stir undue concern or relief but can also lead to unnecessary treatments or investigations.

To limit the occurrence of false positives, scientists set a threshold before research begins鈥攁 significance level, which serves as a benchmark to gauge if results are due to chance or if they reflect a genuine effect. Keeping this level stringent helps in reducing the risk of these pesky Type I errors.

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Most popular questions from this chapter

Flaxseed and Omega-3 Exercise 4.30 on page 271 describes a company that advertises that its milled flaxseed contains, on average, at least \(3800 \mathrm{mg}\) of ALNA, the primary omega-3 fatty acid in flaxseed, per tablespoon. In each case below, which of the standard significance levels, \(1 \%\) or \(5 \%\) or \(10 \%,\) makes the most sense for that situation? (a) The company plans to conduct a test just to double-check that its claim is correct. The company is eager to find evidence that the average amount per tablespoon is greater than 3800 (their alternative hypothesis), and is not really worried about making a mistake. The test is internal to the company and there are unlikely to be any real consequences either way. (b) Suppose, instead, that a consumer organization plans to conduct a test to see if there is evidence against the claim that the product contains at least \(3800 \mathrm{mg}\) per tablespoon. If the organization finds evidence that the advertising claim is false, it will file a lawsuit against the flaxseed company. The organization wants to be very sure that the evidence is strong, since if the company is sued incorrectly, there could be very serious consequences.

A study \(^{20}\) conducted in June 2015 examines ownership of tablet computers by US adults. A random sample of 959 people were surveyed, and we are told that 197 of the 455 men own a tablet and 235 of the 504 women own a tablet. We want to test whether the survey results provide evidence of a difference in the proportion owning a tablet between men and women. (a) State the null and alternative hypotheses, and define the parameters. (b) Give the notation and value of the sample statistic. In the sample, which group has higher tablet ownership: men or women? (c) Use StatKey or other technology to find the pvalue.

Euchre One of the authors and some statistician friends have an ongoing series of Euchre games that will stop when one of the two teams is deemed to be statistically significantly better than the other team. Euchre is a card game and each game results in a win for one team and a loss for the other. Only two teams are competing in this series, which we'll call team A and team B. (a) Define the parameter(s) of interest. (b) What are the null and alternative hypotheses if the goal is to determine if either team is statistically significantly better than the other at winning Euchre? (c) What sample statistic(s) would they need to measure as the games go on? (d) Could the winner be determined after one or two games? Why or why not? (e) Which significance level, \(5 \%\) or \(1 \%,\) will make the game last longer?

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. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Will Vote } \\ \text { Sample A } \end{array} & \text { for Candidate } & \begin{array}{l} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 152 & 98 \\ \text { Flyer } & 145 & 105 \\ \hline \end{array} $$ $$ \begin{array}{lcc} \text { Sample B } & \begin{array}{c} \text { Will Vote } \\ \text { for Candidate } \end{array} & \begin{array}{c} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 188 & 62 \\ \text { Flyer } & 120 & 130 \\ \hline \end{array} $$

Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=28 / 40=0.70\) with \(n=40\)
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