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Chapter 22: Problem 23

The value of the \(z\) statistic for the Exercise \(22.22\) is \(2.53\). This test is (a) not significant at either \(\alpha=0.05\) or \(\alpha=0.01\). (b) significant at \(\alpha=0.05\) but not at \(\alpha=0.01\). (c) significant at both \(\alpha=0.05\) and \(\alpha=0.01\).

Short Answer

Expert verified
(b) significant at \(\alpha=0.05\) but not at \(\alpha=0.01\).

Step by step solution

01

Understand the Given Z-Statistic

The problem states that the value of the \(z\) statistic is \(2.53\). This is a measure which tells us how many standard deviations an element is from the mean in a standard normal distribution.
02

Determine Critical Z-Values

For a Z-test, the critical value for \(\alpha = 0.05\) in a two-tailed test is approximately \(\pm 1.96\), and for \(\alpha = 0.01\), it is approximately \(\pm 2.58\). We will compare the \(z\) statistic with these critical values to determine significance.
03

Evaluate Significance at \(\alpha = 0.05\)

Since the \(z\)-value of \(2.53\) is greater than \(1.96\), it is significant at the \(\alpha = 0.05\) level. This means the probability of observing such an extreme value under the null hypothesis is less than 5%.
04

Evaluate Significance at \(\alpha = 0.01\)

The \(z\)-value of \(2.53\) is less than \(2.58\), meaning it is not significant at the \(\alpha = 0.01\) level. Hence, the probability of observing this value under the null hypothesis is greater than 1%.
05

Conclusion

Given that the \(z\) statistic \(2.53\) is significant at \(\alpha = 0.05\) but not at \(\alpha = 0.01\), the test falls under option (b).

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Key Concepts

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

Understanding the Z Statistic
The Z statistic is a vital part of hypothesis testing, providing a way to measure how far away an observed value is from the mean of a standard normal distribution. Imagine you are dealing with a dataset and you want to know how unusual a specific observation is. The Z statistic helps by expressing this unusualness in terms of standard deviations.
For example, in our problem, the Z statistic is 2.53. This means the observed value is 2.53 standard deviations above the mean.
Once you have the Z statistic, you can compare it to critical Z values to understand if your observation is statistically significant. This step is crucial in testing hypotheses as it shows whether the results of a study or experiment deviate significantly from what is expected under the null hypothesis.
What Significance Level Means
The significance level, denoted by \( \alpha \), is a threshold used to decide whether an observed effect is statistically significant. Common significance levels are 0.05 and 0.01.
When you choose a significance level, you are setting the probability of rejecting the null hypothesis when it's true. For instance, an \( \alpha \) of 0.05 means there is a 5% chance of rejecting the null hypothesis erroneously, known as a Type I error.
  • A significance level of 0.05 is widely used as a conventional threshold in research. It balances between being too lenient and too strict.
  • A significance level of 0.01 offers more stringent criteria, reducing the likelihood of a Type I error but may increase the risk of missing true effects.
In our example, the Z statistic of 2.53 is significant at \( \alpha = 0.05 \) but not at \( \alpha = 0.01 \), showcasing its sensitivity to these thresholds.
The Two-Tailed Test Explained
A two-tailed test in hypothesis testing checks for the possibility of an effect in two directions 鈥� either greater than or less than a certain value. This differs from a one-tailed test, which only considers one direction.
When you conduct a two-tailed test, you are essentially neutral about the direction of the effect. Instead, you're interested in answering a more general question: Is there an effect at all, and if so, is it significantly different from zero (the value under the null hypothesis)?
In our problem:
  • The critical Z values for a two-tailed test at \( \alpha = 0.05 \) are approximately 卤1.96. This range covers both the lower and upper tails.
  • For \( \alpha = 0.01 \), the critical values are approximately 卤2.58, representing a stricter criterion for significance.
In sum, using a two-tailed test here helps to determine whether the Z statistic deviates significantly in either direction from the mean, indicating whether the observed effect is unlikely due to random chance.

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Most popular questions from this chapter

You are planning a survey of students at a large university to determine what proportion favor an increase in student fees to support an expansion of the student newspaper. Using records provided by the registrar, you can select a random sample of students. You will ask each student in the sample whether he or she is in favor of the proposed increase. Your budget will allow a sample of 100 students. (a) For a sample of size 100, construct a table of the margins of error for \(95 \%\) confidence intervals when \(\mathrm{p}^{\wedge \hat{p}}\) takes the values \(0.1,0.3,0.5,0.7\), and \(0.9\). (b) A former editor of the student newspaper offers to provide funds for a sample of size 500 . Repeat the margin of error calculations in part (a) for the larger sample size. Then write a short thank-you note to the former editor describing how the larger sample size will improve the results of the survey.

The proportion of drivers who use seat belts depends on characteristics such as age, sex, and ethnicity. As part of a broader study, investigators observed a random sample of 117 female Hispanic drivers in Boston. Suppose that, in fact, \(60 \%\) of all female Hispanic drivers in the Boston area wear seat belts. In repeated samples, the sample proportion \(p^{\wedge \hat{p}}\) would follow approximately a Normal distribution with mean (a) \(70.2\). (b) \(0.6\) (C) \(0.002\).

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