Question

The heights of 1000 students are approximately normally distributed with a mean of 174.5 centimeters and a standard deviation of 7.9 centimeters. Suppose 200 random samples of size 35 are drawn from this population and the means recorded to the nearest tenth of a centimeter. Determine

(a) the mean and standard deviation of the sampling distribution of X ;

(b) the number of sample means that falls between 172.5 and 175.8 centimeters inclusive;

(c) the number of sample means falling below 172.0 centimeters.

please show all work

Answer 1

Solution :

Given that,

mean = = 174.5

standard deviation = =7.9

a ) n = 35

= 174.5

₌ / n = 7.9  35 = 1.3353

b) P (172.5 < < 175.8 )

P ( 172.5 - 174.5 / 1.3353 ) < ( - / ) < ( 175.8 - 174.5 / 1.3353 )

P ( - 2 / 1.3353< z < 1.3 / 1.3353 )

P (-1.50 < z < 0.97 )

P ( z < 0.97 ) - P ( z < -1.50 )

Using z table

= 0.8340 - 0.0660

= 0.7672

Probability = 0.7672

c ) p ( ,<  172.0 )

P ( - /) < ( 172.0- 174.5 / 1.3353)

P( z < - 2.5 / 1.3353 )

P ( z < -1.87 )   

Using z table

= 0.0307

Probability = 0.0307