Question

100 POINTS please help!!!!

Answer 1

Standard deviation σ = 1.40 (2 d.p.)

Explanation:

If a continuous random variable X is normally distributed with mean μ and variance σ², it is written as:

`\boxed{X \sim\text{N}(\mu,\sigma^2)}`

Given:

• Mean μ = 13.9 g/dL
• P(X < 16.2) = 0.95

Therefore, if the haemoglobin levels are normally distributed:

`X \sim\text{N}(13.9,\sigma^2)`

where X is the haemoglobin level.

Converting to the Z distribution:

`\boxed{\textsf{If }\: X \sim\textsf{N}(\mu,\sigma^2)\:\textsf{ then }\: (X-\mu)/(\sigma)=Z, \quad \textsf{where }\: Z \sim \textsf{N}(0,1)}`

Transform X to Z:

`\text{P}(X < 16.2)= \text{P}\left(Z < (16.2-13.9)/(\sigma)\right)= 0.95`

According to the z-tables, when p = 0.95, z = 1.6449

`\implies (16.2-13.9)/(\sigma)=1.6449`

`\implies (2.3)/(\sigma)=1.6449`

`\implies \sigma=(2.3)/(1.6449)`

`\implies \sigma=1.40`