Homework SourseIf the SAT math scores are normally distributed with a mean
If the SAT math scores are normally distributed with a mean of 520 and a standard deviation of 150.
(a) Find the probability that a randomly selected person has an SAT math score between 490 and 640. (Show work and round the answer to 4 decimal places)
(b) If 225 people are randomly selected, what is the standard deviation of the sample mean SAT math score. (Show work and round the answer to 2 decimal places)
(c) If 225 people are randomly selected, find the probability that sample mean SAT math score is greater than 505. (Show work and round the answer to 4 decimal places)
Solution
Given, Mean = 520 and SD=150
Let X is normal variate with mean 520 and sd 150 and Z is standard normal variate with mean 0 and variance 1
(a) P(490 < X < 640) = P(-0.2 < Z < 0.8)
= f(0.8) - f(-0.2)
= 0.7881 - (1 - f(0.2)) Table values from Area under the Standard Normal Distribution
= 0.7881 - (1-0.5794) Table values from Area under the Standard Normal Distribution
= 0.7881 - 0.4206
= 0.3675
(b) SD(Sample Mean) = SD / Sqrt(n) since sampling distribution of sample mean is SD / sqrt(n)
= 150 / sqrt(225) given Sample size n = 225
= 150 / 15
= 10
(c) P(Sample mean SAT math score is > 505) = P(Z > (505-520)/10)
= P(Z > -1.5)
= 1 - P(Z < -1.5)
= 1 - f (-1.5)
= 1 - ( 1 - f(1.5)) Since f(-z) = 1 - f(z)
= 1 - ( 1 - 0.93319)
=0.9332
