Statistic Question

Question

A population of values has a normal distribution with  and . You intend to draw a random sample of size .

Find the probability that a single randomly selected value is less than 113.1.

P(X < 113.1) = 

Find the probability that a sample of size 

 is randomly selected with a mean less than 113.1.

P ( < 113.1) =

Round answers to 4 decimal places

Answer 1

x is N(110.5,95.6).

Transform x to standard normal distribution (mean 0 SD 1) using z-score for single value:

z = (x - population mean) / SD = (x - 110.5)/95.6

P(x < 113.1) = P(z < (113.1-110.5)/95.6) = P(z < 0.03) = 0.5120 (from standard normal distribution table)

for sample mean, z-sore for mean:

z = (xbar - population mean) / SD/sqrt(sample size) = (xbar - 110.5)/95.6

P(xbar < 113.1) = P(z < (113.1 - 110.5)/(95.6/sqrt(221) = P(z < 0.4) = 0.6554.