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Chapter 22: Problem 23

The value of the \(z\) statistic for the Question \(22.22\) 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
The test is significant at \(\alpha=0.05\) but not at \(\alpha=0.01\).

Step by step solution

01

Understanding the Problem

We need to determine the statistical significance of the given \(z\)-score, which is 2.53. We will compare it with the critical values of the normal distribution for the significance levels \( \alpha = 0.05 \) and \( \alpha = 0.01 \).
02

Identify Critical Values for \( \alpha=0.05 \)

For a two-tailed test at \( \alpha = 0.05 \), the critical \( z \)-values are approximately -1.96 and 1.96. Since 2.53 is greater than 1.96, it is in the rejection region for \( \alpha = 0.05 \).
03

Identify Critical Values for \( \alpha=0.01 \)

For a two-tailed test at \( \alpha = 0.01 \), the critical \( z \)-values are approximately -2.58 and 2.58. Since 2.53 is less than 2.58, it is not in the rejection region for \( \alpha = 0.01 \).
04

Determine Significance

Based on the comparisons, the \( z \)-statistic of 2.53 is significant at \( \alpha = 0.05 \) but not significant at \( \alpha = 0.01 \).

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Key Concepts

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

Z-Score
A Z-Score is a measure that describes a value's relationship to the mean of a group of values. It tells us how many standard deviations a data point is from the mean. If you have a Z-Score of 2.53, it means that the data point is 2.53 standard deviations above the average.
In hypothesis testing, the Z-Score helps us determine how likely it is that our results could have happened by chance. The further away the Z-Score is from zero, the less likely the data point is due to random chance. This is crucial for making decisions about hypotheses. For example, in our exercise, the Z-Score of 2.53 is used to assess whether the data point is in the rejection region for different significance levels.
Statistical Significance
Statistical significance is a determination about whether a result is not likely to happen randomly but rather is likely to be attributable to a specific cause. It helps in understanding whether an observed effect in data or an experiment is real or if it occurred by chance.
  • A statistically significant result means that the observed pattern in the data is strong enough to reject the null hypothesis.
  • In our exercise, the Z-Score of 2.53 was tested for significance, which helps find out if the result is meaningful or not at different alpha levels.
Statistical significance provides confidence that the effect observed in the data is indeed present, rather than merely a result of random variations.
Significance Level
The significance level, often denoted as alpha (伪), is the threshold used in hypothesis testing to decide whether to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true, also known as the Type I error.
Common significance levels used are 0.05 and 0.01. The lower the alpha, the stricter the test is for statistical significance.
  • In the exercise, the Z-Score of 2.53 is compared against critical values for alpha levels of 0.05 and 0.01.
  • Since 2.53 is greater than 1.96, it falls in the rejection region for 伪 = 0.05, making the test significant at this level.
  • However, 2.53 is not greater than 2.58, so it does not reach significance for 伪 = 0.01.
Understanding significance levels helps in correctly interpreting the results of hypothesis testing and determining when evidence is strong enough to reject the null hypothesis.

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Most popular questions from this chapter

Book Reading. Although an increasing share of Americans are reading e-books on tablets and smartphones rather than dedicated e-readers, print books continue to be much more popular than books in digital format (digital format includes both e-books and audio books). A Pew Research Center survey of 1502 adults nationwide conducted January 8-February 7, 2019, found that 1081 of those surveyed had read a book in either print or digital format in the preceding 12 months. 28 (You may regard the 1081 adults in the survey who had read a book in the preceding 12 months as a random sample of readers.) a. What can you say with \(95 \%\) confidence about the percentage of all adults who had read a book in either print or digital format in the preceding 12 months? b. Of the 1081 surveyed who had read a book in the preceding 12 months, 105 had read only digital books. Among those adults who had read a book in the preceding 12 months, find a \(95 \%\) confidence interval for the proportion that had read digital books exclusively.

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No Confidence Interval. The 2017 Youth Risk Behavioral Survey, in a random sample of 497 high school seniors in Connecticut, found that \(0.8 \%\) (that's \(0.008\) as a decimal fraction) smoked cigarettes daily. \({ }^{2}\) Explain why we can't use the large-sample confidence interval to estimate the proportion \(p\) in the population of all Connecticut high school seniors in 2017 who smoked cigarettes daily.

Do You Listen to Podcasts? In January and February 2019, Edison Research conducted a national telephone survey of 1500 Americans aged 12 and older, using random digit dialing techniques to both cell phones and landlines. The survey included questions about the use of mobile devices, Internet audio, podcasting, social media, smart speakers, and more. Of the 1500 people surveyed, 480 said they had listened to a podcast in the past month. 12 a. What is the margin of error of the large-sample \(95 \%\) confidence interval for the proportion of Americans aged 12 and older who had listened to a podcast in the past month? b. How large a sample is needed to get the common \(\pm 3\) percentage point margin of error? Use the January/February survey of 1500 as a pilot study to get \(p^{*}\)

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