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2.2 MORE ON NORMAL DISTRIBUTIONS AND STANDARD NORMAL CALCULATIONS Standardizing We can compare ____________________________ that have different means and standard deviations by standardizing. z x The standardized value is called a ____________________. x is the given value This also allows you to find the __________________________ under a given part of the curve. Why standardizing is your friend Standardizing ________________________________________________________ and your standard deviation to one and compare information in two normal distributions. Who’s Taller? (relatively speaking) Verne is 67” tall. Assume the heights of women her age are normally distributed with a mean μ = 64 inches and standard deviation σ = 2.5 inches. Hank is 72” tall. Assume the heights of men his age are normally distributed with a mean μ = 69.5 inches and standard deviation σ = 2.25 inches. Since any normal curve can be standardized, we can find areas under the curve using one table, Table A. It is very important to remember that Table A gives the __________________________ ______________________________________________________________________!! Also, standardized normal curves have a _____________________________________ ______________________________________________________________________. Use Table A to find the proportion of observations that have a z-score less than 1.4 (this is 1.4 standard deviations from the mean). Use Table A to find the proportion of observations greater than a z-score of -2.15. Steps in Finding Normal Proportions Step 1: Make sure the variable of interest is from a _________________________!!! Then draw a picture of the distribution and _________________________________ ____________________________ with the values given (center and important points). Step 2: Standardize x by using the formula. Label your picture with the __________________________________________. Step 3: Use Table A to _________________________________ under the curve. Step 4: State your conclusion in ____________________________________________ . Now to the actual problems… A commonly used IQ “cut-off” score for AIG identification is 125. IQ scores on the WISC-IV are normally distributed with a mean = 100 and a standard deviation = 15. Find the proportion of people whose IQ score is at least 125. IQs between 140 and 170 are commonly referred to as “moderately profoundly gifted.” What proportion of the population have IQ scores between 140 and 170? Scores on the SAT Verbal approximately follow the N(505,110) distribution. How high must a student score to be in the top 10% of all students taking the SAT? Caution about Test Items Many test items ask students to distinguish between types of density curves. Once the hear the word, students have a tendency to call everything “normal.” Be careful! What if they don’t tell me whether the data are from a normal population? If you’re given the data, you have several ways to assess normality. 1. Start by looking __________________________________________________ _____________________________________________. Does the data appear symmetrical, with most of the data being near the center? 2. And of course there is always the _____________________________… 3. Another method is to check the _______________________ rule. First, find the mean and standard deviation. Then count what percent of the observations fall within one standard deviation of the mean. Is it close to 68%? Repeat for 2 and 3 standard deviations away from the mean. 4. Normal Probability Plots!!!!!! (much easier) i. method is to construct a ___________________________ using your calculator. ii. If the plot is __________________________________, it is safe to assume the data are from a normal distribution. Constructing Normal Prob. Plots Type your data in your calculator (it is probably already there, because I know you have looked at your histogram or box-and-whisker plot!). Go to StatPlot. Choose the last graph option. This represents Normal Probability. Let’s look at page 108, problem 2.26. For those of you who like your calculator… To find the area under a normal curve… 2nd Vars, normal cdf Normalcdf(min, max, μ, σ) If you are looking at everything to the left of the max, then min = 1E-99. If you are looking at everything to the right of the min, then max = 1E99.