NormCalc - Normal Curve Calculations The Empirical Rule...

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Normal Curve Calculations The Empirical Rule that we have applied to bell-shaped distributions is based on a special mathematical distribution called the normal distribution. The normal distribution is symmetric about the mean μ . The standard deviation σ describes the spread from the mean. See figure 1 below. In fact the Empirical Rule says that about 68% of the data falls in the interval μ ± σ and that about 95% falls in the interval μ ± 2 σ . μ μ −σ μ+ σ 514 401 627 740 288 Figure 1 Figure 2 In the year 2000 the SAT math scores had a mean of 514 and a standard deviation of 113. The distribution was approximated well by a normal distribution. The sketch is shown in figure 2 above. By the Empirical Rule we can say that 68% of the SAT math scores were between 401 and 627, and that 95% of the scores were between 288 and 740. What if we wanted to know the percentage of the scores that were between 500 and 700? Or what was the percentage above 700 ? These are the type of questions that we will learn to answer now. The key to answering these questions is to ask how many standard deviations are these scores from the mean. For example how many standard deviations is 700 from the mean of 514 ? (try to answer this before reading further). First 700 is 700 – 514 = 186 points above the mean. How many standard deviations are in 186 points? Answer: 186/113 = 1.65 standard deviations. This is called the standardized score or z-score . The calculation can be summarized by the formula z = x μ σ or z score = value mean s tan dard deviation
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Let’s return to the problem of determining the percentage of the scores that fall between 500 and 700. First calculate the z-scores for each of the scores 500 and 700. The calculations are shown in the sketch in Figure 3. After finding the z-scores we need to determine the percentage of SAT scores between 500 and 700. To do this we use a command in the TI-83 calculator. The command is normalcdf and is found under the DISTR menu which is accessed by 2 nd VARS. Figure 4 shows the shaded area
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This note was uploaded on 10/18/2011 for the course STA 2023 taught by Professor Shaw during the Fall '11 term at Valencia.

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NormCalc - Normal Curve Calculations The Empirical Rule...

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