STAT 400 Lecture AL1 Answers for 7.1, 7.4 Spring 2015 Dalpiaz 1. Suppose the lifetime of a particular brand of light bulbs is normally distributed with standard deviation of = 75 hours and unknown mean. a) What is the probability that in a random sample of 49 bulbs, the average lifetime X is within 21 hours of the overall average lifetime? = 75, n = 49. P( – 21 < X < + 21 ) = 497521Z497521Pμμμμ = P( – 1.96 < Z < 1.96 ) = 0.95. b) Suppose the sample average lifetime of the 49 bulbs is x= 843 hours. Construct a 95% confidence interval for the overall average lifetime for light bulbs of this brand. ( X – 21, X + 21 ) ( 822 , 864 ) A confidence interval is a range of numbers believed to include an unknown population parameter. Associated with the interval is a measure of the confidence we have that the interval does indeed contain the parameter of interest. A (1 ) 100% confidence interval for the population mean when is known and sampling is done from a normal population, or with a large sample, is nznz, 2 2 XXX nz 2 X estimate (point estimate) margin of error = nz 2 1. (continued) Suppose the sample average lifetime of the 49 bulbs is x= 843 hours. b) Construct a 95% confidence interval for the overall average lifetime for light bulbs of this brand. = 75 is known. n = 49 – large. The confidence interval : nσz2αX . 95% confidence level, = 0.05, /2 = 0.025, 2αz = 1.96. 497596.1843 843 21 ( 822 , 864 ) c) Construct a 90% confidence interval for the overall average lifetime for light bulbs. 90% confidence level, = 0.10, /2 = 0.05, 2αz = 1.645. 4975645.1843 843 17.625 ( 825.375 , 860.625 ) d) Construct a 92% confidence interval for the overall average lifetime for light bulbs. 92% confidence level, = 0.08, /2 = 0.04, 2αz = 1.75. 497575.1843 843 18.75 ( 824.25 , 861.75 )Minimum required sample size in estimating the population mean to within with ( 1 – ) 100 % confidence is 2εσz2α n. Always round n up. 2. How many test runs of an automobile are required for determining its average miles-per-gallon rating on the highway to within 0.5 miles per gallon with 95% confidence, if a guess is that the variance of the population of miles per gallon is about 6.25? = 0.5, 2 = 6.25, = 2.5, 95% confidence level, = 0.05, /2 = 0.025, 2αz = 1.960. 222 5.05.296.1αεσzn= 96.04. Round up. n = 97. 1. (continued) e) What is the minimum sample size required if we wish to estimate the overall average lifetime for light bulbs to within 10 hours with 90% confidence? = 10, = 75, 90% confidence level, = 0.10, /2 = 0.05, 2αz = 1.645. 222 1075645.1αεσzn= 152.21390625. Round up. n =
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