Question

On a test of optimism, scores of people diagnosed with a particular disease are normally distributed with a mean of 20 and a standard deviation of 5.

What proportion of the scores are between 20 and 25? Between 25 and 30? Below 15?

Answer 1

`\mathbb P(20<X<25)=\mathbb P\left(\frac{20-20}5<\frac{X-20}5<\frac{25-20}5\right)=\mathbb P(0<Z<1)`

Since approximately 68% of any normal distribution lies within one standard deviation of the mean, and the distribution is symmetric, it follows that


`\mathbb P(0<Z<1)=\frac12\mathbb P(|Z|<1)=\frac{0.68}2=0.34`