Section 7.1 The Standard Normal Curve 293 The TI-84 Plus gives the area as .7190427366. Rounding to four decimal places gives 0.7190. Objective 4 Find z -scores corresponding to areas under the normal curve Finding a z-Score Corresponding to a Given Area Often we are given an area and we need to find the z-score that corresponds to an area under the standard normal curve. Examples 7.7–7.11 show how this is done using either Table A.2 or technology. When using the table, it is useful to remember that the mode, z = 0, has an area of 0.5 both to its right and to its left. EXAMPLE 7.7 Finding the z-score corresponding to an area to the left Explain It Again Finding a z-score corresponding to an area: In Table A.2, the numbers in the body of the table represent areas, and numbers down the left-hand column and across the top row represent z-scores. When given an area, we find that area in the body of the table, and look at the left-hand column and top row to obtain the z-score. Use Table A.2 to find the z-score that has an area of 0.26 to its left. Solution Step 1: Sketch a normal curve and shade in the given area. Step 2: In Table A.2, look through the body of the table to find the area closest to 0.26. This value is 0.2611. The values in the left-hand column and top row corresponding to 0.2611 are −0.6 and 0.04. Therefore, the z-score is z = −0.64. See Figure 7.11. −0.64 Area = 0.26 Figure 7.11 z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ −0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867 −0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148 −0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451 −0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776 −0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ EXAMPLE 7.8 Using technology to find the z-score corresponding to an area to the left In Example 7.7, we found the z-score that has an area of 0.26 to its left. Use technology to find this z-score. Solution We present results from the TI-84 Plus calculator. The z-score is found by using the invNorm command. We enter the area to the left (.26), the mean (0), and the standard deviation (1). Step-by-step instructions are given in the Using Technology section on page 296. The TI-84 Plus gives z = −.6433454021. Rounding to two decimal places gives z = −0.64.
navidi_monk_elementary_statistics_2e_ch7-9
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