Question

Q1. IQ scores are normally distributed with a population mean of 100 and SD 15. If you select a sample of 25 individuals how high would the sample mean have to be to say that probability of getting a sample mean that high or higher is 0.05? Round your answer to whole number.

Q2. 10% of the items produced by a machine are defective. A random sample of 100 items is selected and checked for defects. What is the probability that the sample will contain more than 13% defective units?

Answer 1

a)

X ~ N ( µ = 100 , σ = 15 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.05 = 0.95
To find the value of x
Looking for the probability 0.95 in standard normal table to calculate critical value Z = 1.6449
Z = ( X - µ ) / ( σ / √(n) )
1.6449 = ( X - 100 ) / (15/√(25))
X = 104.935

2)

P( < p) = P(Z < ( - p) / sqrt [ p( 1 - p) / n ]

So,

P( > 0.13) = P(Z > ( 0.13 - 0.10) / sqrt [ 0.10 * ( 1 - 0.10) / 100 ]

= P(Z > 1)

= 0.1587 (From Z table)