Question

Let X represent the weight of the students at a university. Suppose X has a mean of 75 kg and a standard deviation of 10 kg. Among 100 such randomly selected students from this university, what is the approximate probability that the average weight of this sample (X100) lies between

(a) 74 and 75 kg

(b) greater than 76 kg

(c) less than 73 kg

Assume that the sample size(N) is large enough for the CLT (Central Limit Theorem) to be applicable.

Answer 1

Solution :

Given that,

mean = = 75

standard deviation = = 10

n = 100

= = 75

= / n = 10/ 100 = 1

a)

P(74 < < 75) = P((74 - 75) /1 <( - ) / < (75 - 75 ) / 1))

= P(-1 < Z < 0)

= P(Z < 0) - P(Z < -1) Using z table,

= 0.5 - 0.1587

= 0.3413

Probability = 0.3413

b)

P( > 76) = 1 - P( < 76)

= 1 - P(( - ) / < (76 - 75) / 1)

= 1 - P(z < 1)

= 1 - 0.8413 Using standard normal table.

= 0.1587

Probability = 0.1587

c)

P( < 73) = P(( - ) / < (73 - 75) / 1)

= P(z < -2)

= 0.0228 Using standard normal table,   

Probability = 0.0228