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Chapter 17: Problem 26

You are testing \(H_{0}: \mu=0\) against \(H_{\mathrm{a}}: \mu \neq 0\) based on an SRS of 20 observations from a Normal population. What values of the \(z\) statistic are statistically significant at the \(\alpha=0.005\) level? (a) All values for which \(z>2.576\) (b) All values for which \(z>2.807\) (c) All values for which \(|z|>2.807\)

Short Answer

Expert verified
The values for which \(|z| > 2.807\) are statistically significant at \(\alpha = 0.005\), so the correct answer is (c).

Step by step solution

01

Identify the Relevance of the Z-test

The problem statement involves hypothesis testing about the population mean \( \mu \) using a z-test statistic. We are examining if the null hypothesis \( H_0: \mu = 0 \) is to be rejected against an alternative hypothesis \( H_a: \mu eq 0 \).
02

Determine Critical Z-value

For a two-tailed test at significance level \( \alpha = 0.005 \), the critical values of the z-distribution split the 0.005 probability into two tails, each having \( \alpha/2 = 0.0025 \)). We seek the z-values bounding this probability in each tail.
03

Find Z-values from Z-table

Utilize standard normal distribution tables (or a calculator) to find critical z-values that correspond to the cumulative probability of 0.0025 at each tail. This gives \( z \approx 2.807 \) for \( 0.0025 \) in the upper tail and \( z \approx -2.807 \) for \( 0.0025 \) in the lower tail.
04

Identify Significance Range

Since the test is two-tailed, we consider values of \( |z| > 2.807 \) as the critical region. Therefore, both positive and negative z-values beyond \( 2.807 \) in absolute terms are statistically significant at the \( \alpha = 0.005 \) level.

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Key Concepts

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

Z-test
A Z-test is a type of statistical test used to determine if there is a significant difference between sample and population means. It is perfect for situations where the population variance is known, or the sample size is large (typically over 30). The Z-test uses the normal distribution to calculate the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.
This logic helps us decide whether to reject or not reject the null hypothesis. In the context of the given exercise, the Z-test is applied to test whether the population mean (\( \mu \) is equal to zero against the possibility that it is not (two-tailed test).
  • Assumes normal distribution
  • Compares sample mean to population mean
  • Requires known population variance
Significance Level
The significance level, often denoted by \( \alpha \), is a threshold used in hypothesis testing to determine when to reject the null hypothesis. It represents the probability of committing a Type I error, which occurs when the null hypothesis is true but wrongly rejected. A smaller \( \alpha \) indicates a stricter criterion, reducing the likelihood of a Type I error but increasing the possibility of a Type II error (failing to reject a false null hypothesis).
In the exercise, \( \alpha \) is set at 0.005, meaning that there's a 0.5% risk of rejecting the null hypothesis when it is actually true.
  • Controls the probability of Type I error
  • Common alpha values include 0.05, 0.01, and 0.005
  • Lower alpha = stricter hypothesis rejection criteria
Critical Values
Critical values are the threshold values that define the regions where the test statistic leads to the rejection of the null hypothesis. They are determined based on the significance level \( \alpha \) and the distribution of the test statistic. In a Z-test, critical values are derived from the standard normal distribution.
For a two-tailed test with \( \alpha = 0.005\), the area in both tails is \( \alpha/2 = 0.0025\). The critical z-values are symmetric and located at \( z \approx 2.807\) and \( z \approx -2.807\). Any test statistic beyond these limits suggests that the sample observation is unlikely under the null hypothesis.
  • Defines the rejection region
  • Based on significance level and symmetry for two-tailed tests
  • Dependent on test type (z, t, chi-square, etc.)
Two-tailed Test
A two-tailed test is a hypothesis test where the area of interest for extreme values is split between the two tails of the probability distribution. This method tests for the possibility of deviation in both directions - higher or lower. The alternative hypothesis, \( H_a: \mu eq 0\), suggests that we are interested in any significant difference, whether it is positive or negative.
This is why the rejection regions are in both tails of the distribution, characterized by critical values on either side. The given exercise uses a two-tailed test to explore both possibilities of mean deviation from zero.
  • Detects differences in both directions
  • Requires splitting \( \alpha \) into two equal parts
  • Rejection of \( H_0 \) indicates a significant difference

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Most popular questions from this chapter

Every society has its own marks of wealth and prestige. In ancient China, it appears that owning pigs was such a mark. Evidence comes from examining burial sites. The skulls of sacrificed pigs tend to appear along with expensive ornaments, which suggests that the pigs, like the ornaments, signal the wealth and prestige of the person buried. A study of burials from around 3500 B.C. concluded that "there are striking differences in grave goods between burials with pig skulls and burials without them . . . A test indicates that the two samples of total artifacts are statistically significantly different at the \(0.01\) level."8 Explain clearly why "statistically significantly different at the \(0.01\) level" gives good reason to think that there really is a systematic difference between burials that contain pig skulls and those that lack them.

Sketch the standard Normal curve for the \(z\) test statistic and mark off areas under the curve to show why a value of \(z\) that is statistically significant at the \(1 \%\) level in a one-sided test is always statistically significant at the \(5 \%\) level. If \(z\) is statistically significant at the \(5 \%\) level, what can you say about its significance at the \(1 \%\) level?

Athletes performing in bright sunlight often smear black eye grease under their eyes to reduce glare. Does eye grease work? In one study, 16 student subjects took a test of sensitivity to contrast after three hours facing into bright sun, both with and without eye grease. This is a matched pairs design. If eye grease is effective, subjects will be more sensitive to contrast when they use eye grease. Here are the differences in sensitivity, with eye grease minus without eye grease: We want to know whether eye grease increases sensitivity on the average. (a) What are the null and alternative hypotheses? Say in words what mean \(\mu\) your hypotheses concern. (b) Suppose that the subjects are an SRS of all young people with normal vision, that contrast differences follow a Normal distribution in this population, and that the standard deviation of differences is \(\sigma=0.22\). Carry out a test of significance.

You are testing \(H_{0}: \mu=0\) against \(H_{\mathrm{a}}: \mu>0\) based on an SRS of 20 observations from a Normal population. What values of the \(z\) statistic are statistically significant at the \(\alpha=0.005\) level? (a) All values for which \(z>2.576\) (b) All values for which \(z>2.807\) (c) All values for which \(|z|>2.807\)

Research suggests that pressure to perform well can reduce performance on exams. Are there effective strategies to deal with pressure? In an experiment, researchers had students take a test on mathematical skills. The same students were asked to take a second test on the same skills, but now each student was paired with a partner and only if both improved their scores would they receive a monetary reward for participating in the experiment. They were also told that their performance would be videotaped and watched by teachers and students. To help them cope with the pressure, 10 minutes before the second exam they were asked to write as candidly as possible about their thoughts and feelings regarding the exam. "Students who expressed their thoughts before the high-pressure test showed a statistically significant \(5 \%\) math accuracy improvement from the pretest to posttest" \((P<0.03)\). \(^{9} \mathrm{~A}\) colleague who knows no statistics says that an increase of \(5 \%\) isn't a lotmaybe it's just an accident due to natural variation among the students. Explain in simple language how " \(P<0.03\) " answers this objection.
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