314 CHAPTER 7 Estimating Parameters and Determining Sample Sizes The accompanying table lists the most commonly used critical values. Confidence Level A Critical Value, zA,2 90% 0.10 1.645 95% 0.05 1.96 99% 0.01 2.575 The following table shows the relationship between the confidence level and the corresponding value of a. The confidence level of 95% is the value used most often. Most Common Confidence Levels Corresponding Values of A 90% (or 0.90) confidence level: a = 0.10 95% (or 0.95) confidence level: a = 0.05 99% (or 0.99) confidence level: a = 0.01 Critical Value After selecting the confidence level, the associated critical value must be determined. The critical value, sample size, and sample proportion are used to evaluate the margin of error. We will use the following definition of critical value, which was first presented in Section 6-1, and this definition uses a z score from the standard normal distribution. za/2 a/2 a/2 Found from technology or Table A-2 z 5 0 FIGURE 7-1 Critical Value zA,2 in the Standard Normal Distribution DEFINITION For the standard normal distribution, a critical value is a z score on the borderline separating those z scores that are significantly low or significantly high. The number za>2 separates an area of a>2 in the right tail of the standard normal distribution. See Figure 7-2, showing that if a = 0.05, then za>2 = 1.96 (as shown in Example 8 in Section 6-1). Note that when finding the critical z score for a 95% confidence level, we use a cumulative left area of 0.9750 (not 0.95). See Figure 7-2 and think of it this way: Internet Surveys Internet usage among adults has grown from 14% in 1996 to 90% today, and surveys are being dramatically affected by that increase in Internet usage. Internet surveys are faster, cheaper, and enable more efficient analysis of the data collected. Important research has shown that for most types of survey questions, Internet responses are not much different from those obtained through mail or telephone calls. However, some topics, such as those that involve technology, can have a strong bias if respondents are limited to Internet users. Those conducting surveys through the Internet alone should be aware of potential pitfalls. See “Coverage Error in Internet Surveys” provided by the Pew Research Center (pewresearch.org). i Confidence Level: 95% The total area to the left of this boundary is 0.975. a/2 5 0.025 a/2 5 0.025 z 5 0 za/2 51.96 2za/2 521.96 FIGURE 7-2 Finding the Critical Value zA,2 for a 95% Confidence Level This is our The area in both The area in the right The cumulative area from the left, confidence level: tails is: tail is: excluding the right tail, is: 95% u A = 0.05 u A,2 = 0.025 u 1 − 0.025 = 0.975
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