Question

The heights of European 13-year-old boys can be approximated by a normal model with mean μ of 63.6 inches and standard deviation σ of 2.5 inches.

Question 1. What is the probability that a randomly selected 13-year-old boy from Europe is taller than 65 inches?

(use 4 decimal places in your answer)

Question 2. A random sample of 4 European 13-year-old boys is selected. What is the probability that the sample mean height x is greater than 65 inches?

(use 4 decimal places in your answer)

Question 3. A random sample of 9 European 13-year-old boys is selected. What is the probability that the sample mean height x is greater than 65 inches?

(use 4 decimal places in your answer)

Question 4. The Central Limit Theorem was needed to answer questions 1, 2, and 3 above.

True or False?

Answer 1

Solution :

Given that ,

mean = = 63.6

standard deviation = = 2.5

1) P(x > 65) = 1 - p( x< 65)

=1- p P[(x - ) / < (65 - 63.6) / 2.5]

=1- P(z < 0.56)

Using z table,

= 1 - 0.7123

= 0.2877

2) n = 4

= = 63.6

= / n = 2.5/ 4 = 1.25

P( > 65) = 1 - P( < 65)

= 1 - P[( - ) / < (65 - 63.6) / 1.25]

= 1 - P(z < 1.12)

Using z table,    

= 1 - 0.8686

= 0.1314

3) n = 9

= = 63.6

= / n = 2.5/ 9 = 0.833

P( > 65) = 1 - P( < 65)

= 1 - P[( - ) / < (65 - 63.6) / 0.833]

= 1 - P(z < 1.68)

Using z table,    

= 1 - 0.9535

= 0.0.0465

4) False, because sample size is smaller than 30