Notes: 3.2 Standard Normal



Learning Target Use standardized scores to make calculations with normal distributionsWe know how to calculate percentages of values if they are 1, 2, or 3 standard deviations away from the mean. What if they are another distance from the mean…how do we calculate the probabilities?? I can calculate the standardized score (z-score) of a value ?Standardized Score (or Z-Score): Determines the number of standard deviations away from the mean for any value.z= (x-x)st. dev.Positive z-scores are above average and negative z-scores are below average.Example 1: The number of questions correct on a test is normally distributed. The average was 42 with a standard deviation of 4.5. If Johnny got 54 correct, how many standard deviations away from average is he?Sketch and label curve (you can estimate the number of standard deviations by looking)14859007302500To find standard deviations away from the mean, find the z-score: (54-42)4.5 = 2.66Johnny scored 2.66 standard deviations above average.What percent of the students did he do better than? We can’t use the Empirical Rule, so we need to learn a new method.? I can calculate the probability of an event based on standardized scores ?Z-Score Table: Includes z-scores and the area (or probability) that is below the value.-3429006350000Note: You have been given a z-score table to use as a reference. YOU WILL NOT WANT TO LOSE IT!!!To find area under a normal curve (also called proportions, percentile or p-value):Step 1: Find the z-score value in the tableStep 2: Read across the table and find the proportion (% in decimal form) of the area that is below the z-scoreIf you want the area above the value, calculate 1 – proportion.Example 2: Johnny scored 2.66 standard deviations above average. What percent of students did he do better than? What percent of students did better than Johnny?Find z-score of 2.66-457200508000Read down and across the table to find the proportion that corresponds to 2.66.Johnny did better than 0.9961 (99.61% of students scored below him).He did worse than 1 – 0.9961 or 0.0039 of the students (0.39% scored above him)Example 3: Find the p-value of that corresponds to a z-score of 0.19Example 4: Find the percent of values that are above 0.3Example 5: The area under the normal curve that corresponds to z = 0 is 0.5000. Explain why you already knew this.Example 6: Find the proportion of values between 0.35 and 2.4 standard deviations? I can calculate the value of the event based on the standardized score ?To calculate the value of an event based on the z-score, use the formula for z and solve for x.Multiply both sides by the standard deviation (denominator) then add the average to both sides. Example 7: The number of questions correct on a test is normally distributed. The average was 42 with a standard deviation of 5. Jared scored 1.4 standard deviations below average. How questions did he answer correctly on the test?(x-42)5 = - 1.4 (negative because it is below average)x - 42 = - 7x = 35Jared got 35 problems correct on the test.? I can calculate a value of the event based on the probability ?Sometimes we may be given the proportion and want to find the observed value. In this scenario, use Table A backwards:Step 1: Look at all the proportions (decimals) in the table; find the value closest to the given proportion (if the proportion is less than 50% use the negative z side; if it is over 50% use the positive z side)Step 2: Read across and up to find the z score that corresponds to the given proportionStep 3: Plug values into the z-score formula and solve for xExample 8: The number of questions correct on a test is normally distributed. The average was 42 with a standard deviation of 4.5. Mary wants to score at least better than 60% of the students on the test, how many questions must she answer correctly in order to accomplish this?60% = 0.6000 → This value is not in the table so find the closest value → z = 0.25Using the z-score formula: (x-42)4.5 > 0.25Solving for x: x – 42 > 1.125, so x > 43.125Since Mary needs to score at least 43.125, don’t round down…. She needs to get 44 or more correct.Example 9: The average height for females in 64” with a standard deviation of 2.5”. There are 35% of women taller than Judy. How tall is Judy?35% are taller means 65% are shorter (Remember: The values in the table are always below!) → z = 0.39(x-64)2.5 = 0.39so x – 64 = .975x = 64.975 or about 65” (5’5”) ................
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