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Advanced Algebra Name_______________________Unit 6 StatisticsNormal Distribution & Z-Score WorksheetDate_______________Block_____Use the Empirical Rule:A certain brand of automobile tire has a mean life span of 35,000 miles and a standard deviation of 2250 miles. Assume the life spans of the tires have a bell-shaped distribution.Draw a normal distribution that clearly shows the empirical rule.The life spans of the three randomly selected tires are 34,000 miles, 37,000 miles, and 31,000 miles. Find the z-score that corresponds to each life span. According to the z-scores, would the life spans of any of these tires be considered unusual?The life spans of three randomly selected tires are 30,500 miles, 37,250 miles and 35,000 miles. Using the Empirical Rule, find the percentile that corresponds to each life span.What percentage of tires has life spans between 30,500 and 37,250 miles?What percentage of tires has life spans greater than 37,250 miles?The life spans of a species of fruit fly have a bell shaped distribution, with a mean of 33 days and a standard deviation of 4 days.Draw a normal distribution that clearly shows the empirical rule.The life spans of three randomly selected fruit flies are 34 days, 30 days and 42 days. Find the z-score that corresponds to each life span and determine if any of these life spans are unusual.The life spans of three randomly selected fruit flies are 29 days, 41 days and 25 days. Use the Empirical Rule, find the percentile that corresponds to each life span.What percentage of fruit flies lives at most 21 days?What percentage of fruit flies lives longer than 37 days?Intelligence Quotient (IQ) is normally distributed with a mean of 100 and a standard deviation of 15. Sketch a normal distribution curve to reflect this information.Find the probability a person picked at random out of the general population has an IQ in the given interval:Between 100 and 115.ii. Between 85 and 130.Between 130 and 145.iv. Over 130. v. Less than 55. A city’s annual rainfall is approximately normally distributed with a mean of 40 inches and a standard deviation of 6 inches. Use the Empirical Rule to find the probability for each annual rainfall in the city: a. less than 34 inchesb. greater than 46 inches c. greater than 52 inchesd. less than 28 inches e. between 34 and 40 inchesf. between 34 and 46 inches.Use Z Scores:Assume we have a set of data that is normally distributed with a mean of 0 and a standard deviation of 1. If a score were randomly selected find the probability of each of the following:a. P(z > -0.42) b. P(z < 1.28)c. P(z > 1.54) d. P(z < -1.75)e. P(-1.05 < z < 2.6)f. P(-0.09 < z < 1.02) Assume that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. An IQ score is randomly selected from this population. Draw a distribution curve and find the indicated probability.a. P(100<x<127) b. P(x<118)c. P(x>76)Scores on the SAT Verbal test in recent years are approximately normal with a mean of 505 and a standard deviation of 110. How high must the student score in order to place in the top 10% of all students taking the SAT?A college is looking for applicants with SAT Math scores of 750 and above. Assume all scores are normally distributed.In 2004, the scores of males on the SAT Math test had a mean of 537 and a standard deviation of 116. What percent of males scored 750 or above?In the same year, the scores of females on the SAT Math test had a mean of 501 and a standard deviation of 110? What percent of females scored 750 or above?How do the two values compare?Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately normally distributed with a mean of 110 and a standard deviation of 25.a. What percent of people aged 20 to 34 have IQ scores above 100? b. What percent have scores above 150? c. MENSA is an elite organization that admits as members people who score in the top 2% on IQ tests. What score on the Wechsler Adult Intelligence Scale would an individual have to earn to qualify for MENSA membership?The average life of a ‘Neveready’ battery, when used in a TI-83+, is 179 hours with a standard deviation of 37 hours. We have a sample of 900 of the calculators using ‘Neveready’ batteries.a. Approximately how many calculators will have dead batteries after 158 hours of use?b. Approximately how many calculators will still be functioning after 200 hours?c. Approximately how many calculators will stop working between 140 and 175 hours of use? ................
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