{"id":56,"date":"2021-01-12T22:19:35","date_gmt":"2021-01-12T22:19:35","guid":{"rendered":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/using-the-normal-distribution\/"},"modified":"2023-06-26T23:58:20","modified_gmt":"2023-06-26T23:58:20","slug":"using-the-normal-distribution","status":"publish","type":"chapter","link":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/using-the-normal-distribution\/","title":{"rendered":"Using the Normal Distribution"},"content":{"raw":"\r\n[latexpage]\r\n<\/span>\r\n
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The shaded area in the following graph indicates the area to the left of x<\/em>. This area is represented by the probability P<\/em>(X<\/em> < x<\/em>). Normal tables, computers, and calculators provide or calculate the probability P<\/em>(X<\/em> < x<\/em>).<\/p>\r\n\r\n

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\r\n\"This\r\n<\/span><\/figure>\r\n
Figure <\/span>6.4<\/span><\/div>\r\n<\/div>\r\n

The area to the right is then $P(X>x) = 1 \u2013P(X<x)$. Remember, $P(X<x) =$ Area to the left<\/span> of the vertical line through $x$. $P(X >x) = 1 \u2013P(X<x) =$ Area to the right <\/span>of the vertical line through $x$. $P(X<x)$ is the same as $P(X\u2264x)$ and $P(X>x)$ is the same as $P(X\\geq x)$ for continuous distributions.<\/p>\r\n\r\n

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Calculations of Probabilities<\/h3>\r\n

Probabilities are calculated using technology. There are instructions given as necessary for Google Sheets, many of which will work fine in other modern spreadsheet applications as well.<\/p>\r\n\r\n

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NOTE<\/span><\/h3>\r\n<\/header>
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To calculate the probability without a spreadsheet program, use the probability tables provided in the appendix<\/a>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n

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Example <\/span>5.13<\/span><\/h3>\r\n<\/header>
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If the area to the left is 0.0228, then the area to the right is 1 \u2013 0.0228 = 0.9772.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n

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Try It <\/span>5.13<\/span><\/h3>\r\n<\/header>
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If the area to the left of x<\/em> is 0.012, then what is the area to the right?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n

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Google Sheets: Standard Normal Distribution<\/h3>\r\n<\/header>The NORM.S.DIST()<\/a> function assumes $\\mu =0$ and $\\sigma=1$ and will give you the probability\/area to the left of a z-score that you input into the function. For example, if you type =NORM.S.DIST(2.4)<\/strong>, the cell will show the value 0.9918024641 (we generally only need 4 decimal places). This tells us that the probability to the left of $z=2.4$ is 0.9918.\r\n\r\nIn most situations, the mean will not be 0, and\/or the standard deviation will not be 1. In these cases, you can convert the value for which you want the probability (what we sometimes will call the \"$x$-value\") into a $z$-score with the standard formula below and plug that $z$-score into the NORM.S.DIST() function.\r\n\r\n$$z = \\frac{x-\\mu}{\\sigma}$$\r\n

NORM.S.DIST<\/h4>\r\n
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\r\n\r\nReturns the value of the standard normal cumulative distribution function for a specified value.\r\n
Sample Usage<\/h5>\r\nNORM.S.DIST(2.4)<\/code>\r\n\r\nNORM.S.DIST(A2)<\/code>\r\n
Syntax<\/h5>\r\nNORM.S.DIST(x)<\/code>\r\n