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Normal Distributions OBJECTIVE Find the area and probability of a standard normal distribution. RELEVANCE Can find probabilities and values of populations whose data can be represented with a normal distribution. Definition…… Normal Distribution Curve – a symmetric distribution where the data values are evenly distributed about the mean. 2 other names for the normal distribution: a. Guassian b. Bell Curve Skewness…… Three Possible Distributions where the tail indicates skewness: 1. Normal 2. Left (Negative) 3. Right (Positive) Normal…… Normal – mean, median, and mode are all the same. Mean, median, mode Left Skew (Negative)…… Negative Skew – from left to right: mean, median, mode. mean median mode Right Skew (Positive)…… mode median mean Positive Skew- from right to left: mean, median, mode Standard Normal Distributions….. Properties of a Standard Normal Distribution…… 1. 2. 3. 4. 5. Total area under the curve = 1 (or 100%). Mounded and symmetric; Never touches the x-axis. Mean = 0; St. Deviation = 1 The mean divides the area in half → 0.50 on each side. 68% Percentages under the curve: 12stst..dev dev 95% 3 st . dev 99.7% Does this look familiar? 68% 95% 99.7% THE EMPIRICAL RULE! Z-score All values can be transformed from a normal distribution to a standard normal distribution by using the z-score. It represents how many standard deviations “x” is away from the mean. z x Finding area under the curve…… 1. 2. 3. Draw the distribution curve. Shade the area in which you are interested. Use the table to find the areas. You might have to add or subtract or both, depending on your interest. ****The table we are using gives areas from z = 0 to the other z-score. Example……Find the area under the curve between z = 0 and z = 1.52. Look up z = 1.52 The area between z = 0 and z = 1.52 is .4357. You try…Find the area between z = 0 and z = 2.34. Answer: .4904 Find the area between z = 0 and z = -1.75. Answer: .4599 Find the area to the right of z = 1.11 Answer: Half is 0.5. Look up z = 1.11 to get an area of 0.3665. Final Answer: 0.5 – 0.3665 = 0.1335 .3665 x 0 .5 Now you try…… Find the area to the left of z = -1.93. Answer: 0.5 – 0.4732 = 0.0268 Find the area between z = 2.00 and z = 2.47. Look up z = 2.47 to get 0.4932. Look up z = 2 to get 0.4772. Subtract the two areas: 0.4932 – 0.4772 = 0.0160 You try…… Find the area between z = -2.48 and z = -0.83. Answer: 0.4934 – 0.2967 = 0.1967 Find the area between z = -1.37 and z = 1.68. The area for z = 1.68 is 0.4535. The area for z = =1.37 is 0.4147. Add the two areas together for the final answer: 0.4535 + 0.4147 = 0.8682 Find the area to the left of z = 1.99. The area for z = 1.99 is 0.4767. The area for the entire left half of the distribution is 0.5000. Add the two areas together for the final answer: 0.4767 + 0.5000 = 0.9767 Find the area to the right of z = -1.16. The area for z = -1.16 is 0.3770. The area for the entire right half of the distribution is 0.5000. Add the two areas together to get your final answer: 0.3770 + 0.5000 = 0.8770 Find the area to the right of z = 2.43 and to the left of z = -3.01. The area for z = 2.43 is 0.4925. The area for z = -3.01 is 0.4987. Add the two areas together and then subtract from 1: 0.4925 + 0.4987 = 0.9912 1 – 0.9912 = 0.0088 Probability and the Normal Curve Finding Probability Under the Curve…… Use the same procedure as finding the area under the curve; however, there is a different notation. Notations…… Area Notation: a. Between z=0 and z=2.32 b. To the left of z=1.65 c. To the right of z=1.91 d. Between z = -1.2 and z=2.3 Probability Notation: a. P(0 z 2.32) b. P( z 1.65) c. P( z 1.91) d. P(1.2 z 2.3) Find P(0 z 2.32) The area for z = 2.32 is 0.4898. The probability is 0.4898. Find P( z 1.65) The probability for z = 1.65 is 0.4505. The probability for the entire left half of the distribution is 0.5000. The final probability is 0.4505 + 0.5000 = 0.9505. Find P( z 1.91) The probability for z = 1.91 is 0.4719. The probability for the entire right half of the distribution is 0.5000. Subtract the two answers to get your final answer: 0.5000 – 0.4719 = 0.0281. Find P(1.2 z 2.3) The probability for z = -1.2 is 0.3849. The probability for z = 2.3 is 0.4893. Add the two probabilities together to get your final answer: 0.3849 + 0.4893 = 0.8742.