Question

Weights were recorded for all nurses at a particular hospital, the mean weight for an individual nurse was 135 lbs. with a standard deviation of 15. If 19 nurses are selected at random, find the probability that the mean weight is between 125 and 130 lbs

Answer 1

Given:
population mean, μ =135
population standard deviation, σ = 15
sample size, n = 19

Assume a large population, say > 100,
we can reasonably assume a normal distribution, and a relatively small sample.
The use of the generally simpler formula is justified.

Estimate of sample mean

`\bar{x}=\mu=135`

Estimate of sample standard deviation

`\s=\sqrt{(\sigma^2)/(n)}`

`=\sqrt{(15^2)/(19)}=3.44124` to 5 decimal places.

Thus, using the normal probability table,

`P(125<X<130)`

`=P((125-135)/(3.44124)<Z<(130-135)/(3.44124))`

`=P(-2.90593<Z<-1.45297)`

`=P(Z<-2.90593)=0.0018308`

`=P(Z<-1.45297)=0.0731166`

Therefore
The probability that the mean weight is between 125 and 130 lbs
P(125<X<130)=0.0731166-0.0018308
=0.0712858