Question

A random sample of size 30 is selected from a known population with a mean of 13.2 and a standard deviation of 2.1. Samples of the same size are repeatedly collected, allowing a sampling distribution of sample means to be drawn.

a. What is the expected shape of the resulting distribution?

b. Where is the sampling distribution of sample means centered?

c. What is the approximate standard deviation of the sample means?

2. The lifetimes of a certain type of calculator battery are normally distributed. The mean lifetime is 400 days, with a standard deviation of 50 days. For a sample of 6000 new batteries, determine how many batteries will last:

a. between 360 and 460 days.

b. more than 320 days.

c. less than 280 days.

Answer 1

Answer:

2.

Given,

Mean = 400

Standard deviation = 50

Sample = 6000

a)

P(360 < X < 460) = P((360 - 400)/50 < (x-u)/s < (460 - 400)/50)

= P(-0.8 < z < 1.2)

= P(z < 1.2) - P(z < - 0.8)

= 0.8849303 - 0.2118554 [since from z table]

= 0.6731

E(X) = np = 6000*0.6731 = 4038.6 = 4039

b)

P(X > 320) = P((x-u)/s > (320 - 400)/50)

= P(z > -1.6)

= 0.9452007 [since from z table]

= 0.9452

E(X) = 6000*0.9452 = 5671.2 = 5671

c)

P(X < 280) = P(x-u)/s < (280 - 400)/50)

= P(z < -2.4)

= 0.0081975 [since from z table]

= 0.0082

E(X) = 6000*0.0082 = 49.2 = 49