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Research 9 Reading 2 The Normal Distribution Curve The NDC “is a theoretical or ideal mode which was obtained from a mathematical equation” (Levin & Fox, 1990, p 134). The Normal Distribution Curve Continuous probability distribution Used for computing and making statements of probabilities in terms of areas. Used for describing scores (and their distribution) and interpreting the standard deviation. Often called as the “error curve” because random fluctuations about a center occur frequently. The Standard Normal Curve (sources: Gilbert, p 137 & Milton, p 209) 1. Extends indefinitely in both directions but most of the areas under the curve can be found between -3 and +3. 2. The “dips” of the curve, called points of inflection are located 1 standard deviation to either side of the mean. 3. μ = Mdn = Mo Research 9 Reading 2 4. The Standard Normal curve is a probability distribution. The area under the whole curve is 1.0. Learning the Areas under the Normal Curve Use Table 4 (attached to your IM). The entries in this table give the area under the NDC between the mean (z=0) and the given positive value of z. The integral value and the first decimal place of z are given in the stubs at the left; the second decimal place of z is given in the column heads. REMINDER : ALWAYS SKETCH THE AREA NEEDED!!! Examples : (from z to prob.) 1. What is the probability that a z score lies between 0 and 1.45? Pr (0 ≤ Z ≤ 1.45) = Area (z) = 2. What is the probability that a z score lies between 0 and -1.45? Pr (-1.45 ≤ Z ≤ 0) = Area (z) = (negative values of z are not given in the table, however, the standard normal curve is symmetrical) Research 9 Reading 2 3. If a z score is chosen at random, what is the probability that it is at least 1.30 standard deviation units above the mean? Pr (1.30 ≤ z ≤ ά) = = - 4. If a z score is chosen at random, what is the probability that it is at least -1.75? Pr (-1.75 ≤ z ≤ ά) = = + Research 9 Reading 2 5. If a z score is chosen at random, what is the probability that it is at most -1.90? Pr (-ά ≤ z ≤ -1.90) = = - 6. If one score is chosen at random from a population, which is normally distributed with mean of 0 and sd of 1, what is the probability that the score is between .39 and 2.04 standard deviation units from the mean? Pr (.39 ≤ z ≤ 2.04) = = - 7. Determine the proportion of z scores between -1.09 and 2.10. Pr (-1.09 ≤ z ≤ 2.10) = = + Research 9 Reading 2 REVERSE PROBLEMS (from prob. to z) 8. Find the z value if the area under the normal curve is between 0 and A(z) is .4279. OR 9. What is the z score of the 80th percentile? 10. What is the minimum z score of the upper 80 percent of the distribution? Research 9 Reading 2 PROBLEM TYPES Type I problem (from z score to probability/proportion/frequency) (from stub and column head to cell) 1st) Transform the X into a standard score (formula for z score) 2nd) Use the NDC table to determine the appropriate probability/ proportion 3rd) If frequency is required, then multiply the proportion by the total number of cases (e.g. f = prop x N) 4th) Draw your conclusions. Don’t forget to respect the principle of inclusiveness (rounding-UP to the nearest whole number) when calculating frequencies. Type II problem (from probability/proportion/frequency to z score) (from cell to stub and column head) 1st) 2nd) 3rd) 4th) Transform frequency into proportion (e.g. prop = freq / N) Check the cells of the NDC table and choose the nearest value Transform the z value into a raw score (e.g. X = Mean + [Z * SD]) Draw your conclusions. Don’t forget to respect the principle of inclusiveness (rounding-UP to the nearest whole number) when calculating frequencies. CENTRAL LIMIT THEOREM It is used to estimate population parameters and in hypothesis testing. It considers the collection of sample means (sampling distribution of the means) and the standard error of the mean (sd of the sampling distribution of means). Simply put, the CLT uses the mean of a certain sample group (where n is usually 30 or more), instead of using individual values. Thus, the formula is interpreted as: _ Z= X–μ ________ σ / √n Standard error of the mean σ˜x Research 9 Reading 2 APPLICATION (The distribution of the following is assumed to be bell-shaped) 1. The mean of IQ scores is 100 with a standard deviation of 15. If an individual is selected at random, what is the probability that his or her IQ score is a. at least 135? b. between 96 and 119? c. not less than 95? d. between 79 and 92? 2. …if 30 people are randomly selected, what is the probability that their IQ scores are between 96 and 119? 3. Mensa (an organization for people with exceptional IQ) only accepts the upper 1.79% of the population. What should be your minimum IQ to be eligible for Mensa? 4. Given the following information about daily wages: (parameter mean=130Php, SD=25Php, N=500 workers). How MANY workers earn a. at least 125Php? b. Between 125 and 140Php? c. 100Php or less? Liz092015 Research 9 Reading 2