Probability and Statistics Quiz: Normal Distribution and Confidence Intervals, Quizzes of Probability and Statistics

Download Probability and Statistics Quiz: Normal Distribution and Confidence Intervals and more Quizzes Probability and Statistics in PDF only on Docsity! Quiz 5 9/04/08 Name: Directions: Make sure to read each problem carefully. You must show all of your work to receive full credit. Problem 1. Fill in the Blank (2 points): Draw an SRS of size n from any population with mean µ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x̄ is approximately Normally distributed with mean µ and standard deviation σ/ √ n. Problem 2. Young Martians like to play the video game “Earth Invaders”. A particular Martian child, Ned, likes to play the game frequently (though he never gets any better). He marks down all of his scores in a notebook. The scores are Normally distributed with mean µ = 85 and σ = 28. (a) (2 points) If we choose 27 scores at random, what is the probability that the average of these scores is less than 90? P (x̄ < 90) = P ( z < 90− 85 28/ √ 27 ) = P (z < .93) = .8238 (b) (2 points) If we choose 45 scores at random, what is the probability that the sum of the scores is greater than 4100? We restate this in terms of x̄. If the sum is greater than 4100, the average is greater than 4100 45 . P (x̄ > 91.11) = P ( z > 91.11− 85 28/ √ 45 ) = P (z > 1.46) = .0721 (c) (2 points) Suppose we didn’t know the true population mean µ. We collect an SRS of 36 of these scores and get a sample mean of x̄ = 83.5. Create a 98% confidence interval for the true population mean. For 98% confidence, z∗ = 2.326. 83.5± 2.326 28√ 36 = 83.5± 10.85 = (72.65, 94.35)