Question

A distribution of values is normal with a mean of 190 and a standard deviation of 15.

From this distribution, you are drawing samples of size 35. Find the interval containing the middle-most 82% of sample means:

Enter your answer using interval notation. In this context, either inclusive or exclusive intervals would be acceptable. Your numbers should be accurate to 1 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Solution:-

Given that,

mean = = 190

standard deviation = = 15

n = 35

= = 190

= / n = 15 / 35 = 2.54

Using standard normal table,

P( -z < Z < z) = 82%

= P(Z < z) - P(Z <-z ) = 0.82

= 2P(Z < z) - 1 = 0.82

= 2P(Z < z) = 1 + 0.82

= P(Z < z) = 1.82 / 2

= P(Z < z) = 0.91

= P(Z < 1.341) = 0.91

= z  ± 1.341

Using z-score formula  

= z * +   

= -1.341 * 2.54 + 190

= 186.6

Using z-score formula  

= z * +   

= 1.341 * 2.54 + 190

= 193.4

( 186.6, 193.4 )