Question

The mean of a population is 77 and the standard deviation is 12. The shape of the population is unknown. Determine the probability of each of the following occurring from this population. a. A random sample of size 35 yielding a sample mean of 81 or more

b. A random sample of size 150 yielding a sample mean of between 76 and 80

c. A random sample of size 221 yielding a sample mean of less than 77.2

(Round all the values of z to 2 decimal places and final answers to 4 decimal places.)

Answer 1

Solution :

Given that,

mean = = 77

standard deviation = = 12

a )n = 35

=  77

₌ / n = 12 35 = 2.0284

P ( > 81 )

= 1 - P ( < 81 )

= 1 - P ( - /) < (81 - 77 /2.0284)

= 1 - P( z < 4 / 2.0284)

= 1 - P ( z < 1.97 )   

Using z table

= 1 - 0.9756

= 0.0244

Probability = 0.0244

b )n = 150

=  77

₌ / n = 12 150 = 0.9798

P 76< < 80 )

P ( 76  - 77/ 0.9798) < ( - / ) < ( 80 - 77 / 0.9798)

P ( - 1 / 0.9798< z < 3 /0.9798 )

P (-1.02 < z < 3.06 )

P ( z < 3.06 ) - P ( z < -1.02)

Using z table

=0.9989 - 0.1539

= 0.8450

Probability = 0.8450

c )n = 221

=  77

₌ / n = 12 221 = 0.8072

P ( < 77.2 )

P ( - /) < (77.2 - 77 /0.8072)

P( z < 0.2 / 0.8072)

P ( z < 0.25 )   

Using z table

=0.5987

Probability = 0.5987