Introduction: I'm studying "Statistics-Business-Economics-Paul-Newbold", chapter 6, topic: "Acceptance intervals" (page: 260).


How Newbold defines acceptance intervals:(in other words, briefly) Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter (mu). Then, Newbold explains the formula below and says that alpha (the probability that sample mean is out of acceptance region), normally, is (alpha < 0.01) and that (z_alpha/2 = 3).

Problem: I can't understand/visualize/imagine the real meaning of ($z_{\alpha/2}$) in the book's notation:

$$\left( \bar{x}-z_{\alpha/2}\frac{\sigma}{\sqrt{n}},\bar{x}+z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right)$$

Specific question: How can I describe z (also called z_alpha/2 in Newbold) in the formula of acceptance intervals?

  • 1
    $\begingroup$ As far as possible, please make your question stand alone. Links are good for support, but the question should offer context. Please explain how Newbold defines acceptance intervals (with a short direct quote if possible), and then please ask a specific question in your body text. Your points 1-3 at the end are unclear/confusing. Your shotgun title "interpret/explain/understand" is too broad. $\endgroup$ – Glen_b Sep 1 '17 at 9:42
  • $\begingroup$ I'm still not 100% clear on the point of acceptance intervals here. Is Newman simply defining the complement of a rejection region for use in a hypothesis test or is he creating a consonance interval (an interval for a parameter based on the region in which some test would not reject)? $\endgroup$ – Glen_b Sep 1 '17 at 10:43
  • $\begingroup$ The second option. I'm studying the basics. The main idea in the book: acceptance interval (=confidence interval)=the interval in which, if I already know mu (mean) and variance (squared sigma) of the population, the sample mean (x̄ uppercase) has excellent probability to find (to stay) in the interval. Thanks. $\endgroup$ – user175690 Sep 1 '17 at 11:08
  • $\begingroup$ @Glen_b It is about confidence intervals. $z$ is the standard normal random variable, $z_{\alpha/2}$ is the $\alpha/2$ quantile of its distribution. $\endgroup$ – user83346 Sep 1 '17 at 11:49
  1. $+- z_{\alpha/2}$ == critical values;
  2. $(1 - {\alpha})$ == confidence level;
  3. common confidence levels: 0.90, 0.95 and 0.99 (or 90%, 95% and 99%);
  4. confidence level 90%: $z_{\alpha/2}$ = $z_{\ 0.05}$ = 1.645

    confidence level 95%: $z_{\alpha/2}$ = $z_{\ 0.025}$ = 1.96

    confidence level 99%: $z_{\alpha/2}$ = $z_{\ 0.005}$ = 2.576;

  5. Links: http://www.dummies.com/education/math/statistics/checking-out-statistical-confidence-interval-critical-values/



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