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Normal Probability Distribution with TI 83/84 calculator The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use this to graph a normal curve. Using this function returns the y-coordinates of the normal curve. Syntax: normalpdf (x, mean, standard deviation) #2: normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set the lower bound. Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) #3: invNorm( inv = Inverse Normal Probability Distribution Function This function returns the x-value given the probability region to the left of the xvalue. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Syntax: invNorm (probability, mean, standard deviation) Example 1: Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is between 65 and 80, inclusive. b) the probability that a value is greater than or equal to 75. c) the probability that a value is less than 62. d) the 90th percentile for this distribution. (answers will be rounded to the nearest thousandth) 1a: Find the probability that a value is between 65 and 80, inclusive. (This is accomplished by finding the probability of the cumulative interval from 65 to 80.) Syntax:normalcdf(lower bound, upper bound, mean, standard deviation) Answer: The probability is 1b: Find the probability that a value is greater than or equal to 75. (The upper boundary in this problem will be positive infinity. The largest value the calculator can handle is 1 x 1099. Type 1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) Answer: The probability is 1c: Find the probability that a value is less than 62. (The lower boundary in this problem will be negative infinity. The smallest value the calculator can handle is -1 x 1099. Type -1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) Answer: The probability is 1d: Find the 90th percentile for this distribution. (Given a probability region to the left of a value (i.e., a percentile), determine the value using invNorm.) Answer: The x-value is Example 2: Graph and examine a situation where the mean score is 46 and the standard deviation is 8.5 for a normally distributed set of data. Go to Y= . 1. Adjust the window. GRAPH. Practice and Homework (Independenteven # and HW# odd) The amount of mustard dispensed from a machine at The Hotdog Emporium is normally distributed with a mean of 0.9 ounce and a standard deviation of 0.1 ounce. If the machine is used 500 times, approximately how many times will it be expected to dispense 1 or more ounces of mustard. Choose: 5 16 80 100 2. Professor Halen has 184 students in his college mathematics lecture class. The scores on the midterm exam are normally distributed with a mean of 72.3 and a standard deviation of 8.9. How many students in the class can be expected to receive a score between 82 and 90? Express answer to the nearest student. 3. A machine is used to fill soda bottles. The amount of soda dispensed into each bottle varies slightly. Suppose the amount of soda dispensed into the bottles is normally distributed. If at least 99% of the bottles must have between 585 and 595 milliliters of soda, find the greatest standard deviation, to the nearest hundredth, that can be allowed. 4. Residents of upstate New York are accustomed to large amounts of snow with snowfalls often exceeding 6 inches in one day. In one city, such snowfalls were recorded for two seasons and are as follows (in inches): 8.6, 9.5, 14.1, 11.5, 7.0, 8.4, 9.0, 6.7, 21.5, 7.7, 6.8, 6.1, 8.5, 14.4, 6.1, 8.0, 9.2, 7.1 What are the mean and the population standard deviation for this data, to the nearest hundredth? 5. Neesha's scores in Chemistry this semester were rather inconsistent: 100, 85, 55, 95, 75, 100. For this population, how many scores are within one standard deviation of the mean? 6. Battery lifetime is normally distributed for large samples. The mean lifetime is 500 days and the standard deviation is 61 days. To the nearest percent, what percent of batteries have lifetimes longer than 561 days? 7. The number of children of each of the first 41 United States presidents is given in the accompanying table. For this population, determine the mean and the standard deviation to the nearest tenth. How many of these presidents fall within one standard deviation of the mean? 8. From 1984 to 1995, the winning scores for a golf tournament were 276, 279, 279, 277, 278, 278, 280, 282, 285, 272, 279, and 278. Using the standard deviation for this sample, Sx, find the percent of these winning scores that fall within one standard deviation of the mean. 9. A shoe manufacturer collected data regarding men's shoe sizes and found that the distribution of sizes exactly fits the normal curve. If the mean shoe size is 11 and the standard deviation is 1.5, find: a. the probability that a man's shoe size is greater than or equal to 11. b. the probability that a man's shoe size is greater than or equal to 12.5. c. 10. Five hundred values are normally distributed with a mean of 125 and a standard deviation of 10. a. What percent of the values lies in the interval 115 - 135, to the nearest percent? b. What percent of the values is in the interval 100 - 150, to the nearest percent? c. What interval about the mean includes 95% of the data? d. What interval about the mean includes 50% of the data? 11. A group of 625 students has a mean age of 15.8 years with a standard deviation of 0.6 years. The ages are normally distributed. How many students are younger than 16.2 years? Express answer to the nearest student?