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Determining Probabilities Under the Normal Curve So far we have looked at boundaries for shaded regions under the normal curves; they have been exactly one or two standard deviations away from the mean. However, an individual’s z-score in any kind of continuous probability distribution can equal a decimal value that is between or beyond these whole zscores. So to work with this, we need to use more ways to find the probabilities within the given range of scores. All the values within the z-score table represent the area under the curve between the mean of the distribution and the given z-scores. Since this is a z-score table as opposed to an actual data table, the mean of the data is assumed to be zero, while the standard deviation is 1. Positive Z-scores Use the z-score table to calculate the probability of the indicated range of z-scores. (a) p(0<z<1.2) (b) p(z<1.2) (c) p(z<1.85) (d) p(z>0.73) (e) p(0.5<z<2.0) (f) p(z=0.33) Negative Z-scores Negative scores are below the mean. Example p(z<-0.67) p(z>0.67) Example Use the z-score table to calculate the probability of the indicated range of z-scores. (a) p(-0.4<z<0) (b) p(z<-0.4) (c) p(z>-0.43) (d) p(-0.55<z<1.55) (e) p(-0.67<z<0.67) (f) (-1.8<z<0) Revisiting Z-score Calculations Example Determine Michael’s z-score on a physical examination if he scored 70 out 100, where the mean of the population is 66 with a standard deviation of 8. Example Find the mean of a set of data with a standard deviation of 12, given that a value of 74 has a z-score of 1.5. Raw Data and the Z-score Table In many situations, data is not presented as a z-score, but as raw values relevant to a given situation. For instance, the scores on an exam might be normally distributed with a mean of 75 and a standard deviation of 9. You must convert any raw data into z-scores before you can carry out any probability analyses using the z-score table. Example Find the probabilities for the following X values, given the mean and standard deviation of the data set: (a) Given μ= 5 and δ = 2, find p(X<4.5) (b) Given μ= 2 and δ = 0.2, find p(X>1.5) (c) Given μ= 50 and δ = 10, find p(X<55) Real-world Applications of the Normal Distribution Example The life span of a particular species of turtles is normally distributed with a mean of 180 years and a standard deviation of 40 years. (a) What is the probability that a randomly chosen turtle will live for less than 150 year? (b) What is the probability that a randomly chosen turtle will live for more than 230 years? (c) What is the probability that a randomly chose turtle will live between 160 and 210 years? Percentiles The term percentile is used, often incorrectly. For example, a person who is extremely tall might said to be in the top 1% of all people, according to height. However, this does not mean that he or she is in the “first percentile.” Instead, the person is said to be in the 99th percentile, because this measure indicates by how many individuals the person is ahead of a particular variable. Therefore, for favourable variables such as intelligence and happiness, it is good to be in a higher percentile. However, it is better to be in a lower percentile for less favourable variables, such as being susceptible to a particular disease. Example On a college entrance exam, the scores are normally distributed. The mean is 245 and the standard deviation is 40. (a) If Fiona scored 300, what percentile is he in? (b) If Jesse scored 220, what percentile is he in?