{"id":101,"date":"2018-10-31T17:43:55","date_gmt":"2018-10-31T21:43:55","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/simplestats\/?post_type=chapter&p=101"},"modified":"2019-10-18T18:55:17","modified_gmt":"2019-10-18T22:55:17","slug":"6-7-confidence-intervals","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/simplestats\/chapter\/6-7-confidence-intervals\/","title":{"raw":"6.7 Confidence Intervals","rendered":"6.7 Confidence Intervals"},"content":{"raw":"[latexpage]\r\n

In our discussion on statistical inference so far, I have used not one type of estimators but two<\/em>, without bringing your attention to it. Probability theory and the Central Limit Theorem describing the sampling distribution of statistics provide us with two types of estimators, called point estimators<\/em> and interval estimators<\/em>.<\/p>\r\n \r\n\r\nA single sample statistic which estimates a population parameter<\/strong> -- and which offers a \"best guess\" for that parameter -- is a\u00a0point<\/em> estimator.\u00a0<\/strong>We have worked with several point estimates by now: the sample mean $\\overline{x}$ is a point estimate of the population mean \u03bc <\/em>while\u00a0the sample standard deviation s<\/em> is a point estimate (which we can note as $\\hat{\\sigma}$) of the population standard deviation\u00a0\u03c3<\/em>.\r\n\r\n \r\n\r\nSimilarly, I'll add another useful point estimate, of the sample proportion<\/em>. Imagine we are interested in studying unemployment. We take a random sample which reveals that, say, 10 percent of the sample respondents report being unemployed. Thus, we have the sample proportion p<\/em>\u00a0as 0.1 and we can use that proportion as a point estimate of the proportion of the population which is unemployed. We denote population proportions by the small-case Greek letter for\u00a0p\u00a0<\/em>which is\u00a0\u03c0<\/em>[footnote]Pronounced PAI, as you probably already know from the mathematical constant \u03c0<\/em>=3.14. While we use the letter \u03c0<\/em> for both population proportions and the mathematical constant, context provides enough clues to differentiate them.[\/footnote]. In other words, the sample proportion p<\/em> serves as a point estimate of the population proportion\u00a0\u03c0<\/em>.\r\n\r\n \r\n\r\nYou'll be happy to know that you are also already familiar with the other, interval<\/em>, type of statistical estimators. As their name suggests, interval estimators, called confidence intervals<\/em>, provide not just one number as a best guess but a whole set of plausible values for the population parameter.<\/strong>\r\n\r\n \r\n\r\nIf you recall Examples 6.3 and 6.4 from the previous section, you'll recognize that we already calculated confidence intervals. In Example 6.4 on the average annual salary, we found\u00a0a range of values within which the average annual salary of the city population was estimated to fall. Specifically, the average annual salary of the random sample was \\$50,000 and we were able to estimate\u00a0with 95% certainty\u00a0<\/span>that the average annual salary of the city population would fall between \\$49,400 and \\$50,600. This range of values between \\$49,400 and \\$50,600 is in effect a confidence interval (a 95% confidence interval, to be precise). The actual numbers \"bracketing\" the interval are called error bounds<\/em>; the interval itself is between, and including, the\u00a0lower error bound<\/em> and the\u00a0upper error bound.\u00a0<\/em><\/span>\r\n\r\n \r\n\r\nUp until now, we calculated the confidence interval in a fast and easy way as I wanted to get the point of the logic underlying statistical inference across. At this time, however, we need to get more technical and precise about it.\r\n\r\n \r\n\r\nFirst, let's revisit how we did it in the previous section to refresh your memory; then I'll show you the more<\/em> correct way to do it. (Before you panic, know that what we did before was not incorrect; we just used rounded numbers to make calculations easier\/faster.)\r\n\r\n \r\n\r\nThis is the information about the sample mean and standard deviation we had from Example 6.4\u00a0Average Annual Income\u00a0<\/em>(without the dollar signs for clarity of presentation):\r\n\r\n \r\n\r\n$\\overline{x}=50000$\r\n\r\n$s=12000$\r\n\r\n$N=1600$\r\n\r\n \r\n\r\nOur starting point is the sample mean (which, according to the CLT approximates the population mean, with large N<\/em>). In order to calculate a confidence interval around the sample mean $\\overline{x}$, we first need to get the standard error $\\sigma_\\overline{x}$, given by the CLT-based formula:\r\n\r\n \r\n\r\n$\\sigma_\\overline{x}$ $=\\frac{\\sigma}{\\sqrt{N}}$\r\n\r\n \r\n\r\nWe don't know the population standard deviation $\\sigma$ but we estimate it with its point estimator s<\/em>, so we get:\r\n\r\n \r\n\r\n$\\hat\\sigma_\\overline{x}$ $=s_\\overline{x}$ $=\\frac{s}{\\sqrt{N}}$\r\n\r\n \r\n\r\nSubstituting s<\/em> and N<\/em> in the formula gives us the following:\r\n\r\n \r\n\r\n$s_\\overline{x}$ $=\\frac{12000}{\\sqrt{1600}}=\\frac{12000}{40}=300$\r\n\r\n \r\n\r\nNow, by the CLT, we have everything we need for the sampling distribution: its mean (as estimated by the sample mean $\\overline{x}$, its standard deviation (i.e., the standard error $\\sigma_\\overline{x}$), and its shape as a normal curve. From Section 5.2.4 (https:\/\/pressbooks.bccampus.ca\/simplestats\/chapter\/5-2-4-the-real-normal-distribution\/<\/a>), we know the probabilities under the normal curve, and that 68 percent of cases[footnote]Here you can imagine the cases as the hypothetical means over infinite sampling.[\/footnote] fall within 1 standard deviation from the mean while 95 percent of cases fall within about 2 standard deviations from the mean.\r\n\r\n \r\n\r\nThe resulting graph was presented in Fig. 6.2 in the previous section. Here it is again, this time with the 68 percent demarcations included.\r\n\r\n \r\n\r\nFigure 6.3\u00a0Average Annual Income (in thousands of dollars), Revisited\u00a0<\/em>\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nHowever, to calculate a confidence interval we don't need to draw the sampling distribution every time; we just need to keep in mind what it represents in terms of probabilities.\r\n\r\n \r\n\r\nFrom our discussion of Example 6.4 in the previous section and now, we can easily deduce the basic formula for calculating a confidence interval:\r\n