Applications of Normal Distribution in Statistical Analysis

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Applications of Normal Distribution in Statistical AnalysisWeek 48.Assume the speed of vehicles along a stretch of I-10 has an approximately normaldistribution with a mean of 71 mph and a standard deviation of 8 mph.a.The current speed limit is 65 mph. What is the proportion of vehicles less than orequalto the speed limit?b.What proportion of the vehicles would be going less than 50 mph?c.A new speed limit will be initiated such that approximately 10% of vehicles willbe overthe speed limit. What is the new speed limit based on this criterion?d. In what way do you think the actual distribution of speeds differs from a normaldistribution?Ans.a.= 71 mph;= 8 mphThe proportion of vehicles less than or equal to the speed limit 65 mph = 0.2266b. The proportion of the vehicles would be going less than 50 mph = 0.0043c. The new speed limit would be 81.25 mph (approximately 81 mph).d. The actual distribution of speeds on a highway differs from thenormaldistribution in thati. It is not symmetric. The speeds are not symmetrically distributed about thecentralvaue. Two speeds that are an equalamount greater or lowerthan the mean speed,may not have the same frequency as is required for a normal distribution.ii. Secondly, the speeds further away from the average speed do not drop infrequency as rapidly as in normal distribution. The frequency of the extreme valuesmay not be as low as in normal distribution, resulting in a fatter tail. Thedistribution may also be skewed in one direction. This is specially true in highwayswhere the distribution will be expected to be positively skewed-greater frequencyin the higher speed range than lower.11.A group of students at a school takes a history test. The distribution is normal witha mean of 25, and a standard deviation of 4. (a) Everyone who scores in the top 30%of the distribution gets a certificate. What is the lowest score someone can get andstill earn a certificate? (b) The top 5% of the scores get to compete in a statewidehistory contest. What is the lowest score someone can get and still go onto competewith the rest of the state?

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Ans.For the test, the mean score= 25;= 4Using the normal calculator (Ch-7, Lane),a. the lowest score someone can get and still earn a certificate = 27.096 (or approx..27.1)b. the lowest score someone can get and still go onto compete with the rest of thestate = 31.58 (or approx. 31.6)12.Use the normal distribution to approximate the binomial distribution and find theprobability of getting 15 to 18 heads out of 25 flips. Compare this to what you getwhen you calculate the probability using the binomial distribution. Write youranswers out to four decimal places.AnsForthe binomial distributionof coin flipping, probability of getting a head(success)in one flip =p=0.5N = no. of trials = 25Mean is= N*p = 25 * 0.5 =12.5Varianceσ2= Np(1-p) = (25)(0.5)(0.5) = 6.25The standard deviation is thereforeσ=√6.25 = 2.5.Using the binomial calculator (Lane CH-5),theprobability of getting 15 to 18 heads out of 25 flips = 0.2049Using thenormal distributionto approximate the binomial distribution:Thearea between 14.5 and 18.5in the normal distribution curveis an approximation ofthe probability of obtaining 15 to 18 heads.Area below14.5 = 0.7881Area below 18.5 = 0.9918Therefore,Area between 14.5 and 18.5 = 0.9918–0.7881 = 0.2037So in this case,theprobability of getting 15 to 18 heads out of 25 flips = 0.2037

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