Question

A population has a mean of 400 and a standard deviation of 50. Suppose a sample of size 125 is selected and  is used to estimate . Use z-table.

1. What is the probability that the sample mean will be within +/- 9 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)
     
   
2. What is the probability that the sample mean will be within +/- 10 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)
   

Answer 1

Solution :

Given that ,

a) mean = = 400

standard deviation = = 50

n = 125

=   =400

= / n =50/ 125 = 4.4721

a ) within 10 = 400  400± 10 = 390, 410

P(390 < < 410)  

= P[(390 - 400) / 4.4721 < ( - ) / < (410 - 400) /4.4721 )]

= P(-2.24 < Z < 2.24)

= P(Z < 2.24) - P(Z < -2.24 )

Using z table,  

= 0.9875 - 0.0125

= 0.9750

Probability = 0.9750

b ) within 9 = 400 ± 9 = 391, 409

P(391 < < 409)  

= P[(391 - 400) / 4.4721 < ( - ) / < (409 - 400) /4.4721 )]

= P(-2.01 < Z < 2.01)

= P(Z < 2.01) - P(Z < -2.01 )

Using z table,  

= 0.9778 - 0.0222

= 0.9556

Probability = 0.9556