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Chapter 9: Problem 43

Error probability A significance test about a proportion is conducted using a significance level of \(0.05 .\) The test statistic equals \(2.58 .\) The P-value is 0.01 a. If \(\mathrm{H}_{0}\) were true, for what probability of a Type I error was the test designed? b. If this test resulted in a decision error, what type of error was it?

Short Answer

Expert verified
a. The test was designed with a Type I error probability of 0.05. b. The error was a Type I error.

Step by step solution

01

Understanding Type I Error

A Type I error occurs when the null hypothesis \( H_0 \) is true, but we mistakenly reject it. The probability of committing a Type I error is denoted by the significance level \( \alpha \). In this test, \( \alpha = 0.05 \).
02

Identifying Test Design

The test was designed at a significance level of \( 0.05 \). This means if \( H_0 \) were true, the probability of making a Type I error, or rejecting \( H_0 \) incorrectly, would be \( 0.05 \).
03

Comparing Test Statistic and Critical Value

The given test statistic is \( 2.58 \). For a significance level of \( 0.05 \) in a two-tailed test, the critical z-values are approximately \(-1.96\) and \(1.96\). Since \( 2.58 \) exceeds \( 1.96 \, \) it falls in the rejection region, leading to the rejection of \( H_0 \).
04

Understanding Decision Error Type

If the test results in a decision error, since we rejected \( H_0 \) and \( H_0 \) was true, this error is indeed a Type I error. A Type I error is rejecting \( H_0 \) when it is actually true.

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Key Concepts

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

Type I Error
In hypothesis testing, committing a Type I error means rejecting the null hypothesis \( H_0 \) when in fact it is true. This is often referred to as a "false positive". The significance level \( \alpha \), such as 0.05 in this case, represents the probability of making such an error.
For example, if you set \( \alpha = 0.05 \), you are accepting a 5% risk of rejecting \( H_0 \) erroneously. This scenario might occur when testing a new drug and concluding it works when it actually doesn't. Therefore, reducing the Type I error probability is crucial in ensuring the validity of test results.
P-value
A P-value is a crucial component in significance testing. It measures the probability of obtaining test results at least as extreme as the observed data, under the assumption that the null hypothesis \( H_0 \) is true.
In essence, the P-value helps us gauge the strength of evidence against \( H_0 \). A lower P-value, such as 0.01 as indicated in the exercise, suggests stronger evidence against the null hypothesis. It means there is only a 1% chance that the observed outcome would occur if \( H_0 \) were true. This supports the decision to reject \( H_0 \) with confidence.
Null Hypothesis Evaluation
Evaluating the null hypothesis \( H_0 \) requires understanding its role in significance testing. The null hypothesis typically represents a default position or a statement of no effect. When performing tests, we start by assuming \( H_0 \) is true.
Our goal is to determine whether there is enough evidence to reject it in favor of the alternative hypothesis \( H_1 \). This is done by analyzing P-values and test statistics. If the P-value is lower than \( \alpha \), we reject \( H_0 \). In the exercise, because the P-value of 0.01 is lower than the significance level of 0.05, \( H_0 \) is rejected, showing evidence in favor of an effect.
Critical Value
A critical value serves as a threshold to determine whether to reject the null hypothesis \( H_0 \). It marks the boundary beyond which observed data is considered statistically significant. Critical values depend on the significance level \( \alpha \) and the chosen test, one-tailed or two-tailed.
For a significance level of 0.05 in a two-tailed test, the critical z-values are roughly \(-1.96\) and \(1.96\). If the test statistic exceeds these critical z-values, as with the test statistic of 2.58 in the exercise, it indicates the data lies in the rejection region. Thus, we reject \( H_0 \), suggesting a statistically significant result.

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Most popular questions from this chapter

Young workers When the 127 workers aged \(18-25\) in the 2008 GSS were asked how many hours they worked in the previous week, the mean was 37.47 with a standard deviation of \(13.63 .\) Does this suggest that the population mean work week for this age group differs from 40 hours? Answer by: a. Identifying the relevant variable and parameter. b. Stating null and alternative hypotheses. c. Finding and interpreting the test statistic value. d. Reporting and interpreting the P-value and stating the conclusion in context.

Astrology errors Example 3 , in testing \(\mathrm{H}_{0}: p=1 / 3\) against \(\mathrm{H}_{a}: p>1 / 3,\) analyzed whether astrologers could predict the correct personality chart (out of three possible ones) for a given horoscope better than by random guessing. In the words of that example, what would be (a) a Type I error and (b) a Type II error?

Test and CI Results of \(99 \%\) confidence intervals are consistent with results of two-sided tests with which significance level? Explain the connection.

Errors in medicine Consider the test of \(\mathrm{H}_{0}\) : The new drug is safe against \(\mathrm{H}_{a}\) : the new drug is not safe. a. Explain in context the conclusion of the test if \(\mathrm{H}_{0}\) is rejected. b. Describe the consequence of a Type I error. c. Explain in context the conclusion of the test if you fail to reject \(\mathrm{H}_{0}\) d. Describe the consequence of a Type II error.

Find test statistic and P-value For a test of \(\mathrm{H}_{0}: p=0.50\), the sample proportion is 0.35 based on a sample size of 100 . a. Show that the test statistic is \(z=-3.0\). b. Find the \(\mathrm{P}\) -value for \(\mathrm{H}_{a}: p<0.50\). c. Does the P-value in part b give much evidence against \(\mathrm{H}_{0}\) ? Explain.
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