Find P(X < 40.851) where X follows a normal distribution with mean 55 and standard deviation 10.

Step-by-step solution

1. 1

   Standardise the value into a $z$-score. The $z$-score measures how many standard deviations the value sits from the mean:

   $z=\frac{x-\mu}{\sigma}=\frac{40.851-55}{10}=\frac{-14.149}{10}=-1.4149$

   The negative sign says $40.851$ is below the mean, so the probability must come out below $0.5$ — an immediate check on the final answer.

2. 2

   Translate the question into a standard-normal area. Because standardising preserves probabilities,

   $P(X<40.851)=P(Z<-1.4149)=\Phi(-1.4149)$

   where $\Phi$ is the standard normal cumulative distribution function.

3. 3

   Evaluate the cumulative probability. Using the error function, $\Phi(z)=\tfrac12\left[1+\operatorname{erf}\!\left(\tfrac{z}{\sqrt2}\right)\right]$:

   $\Phi(-1.4149)=0.078549$

   From a printed table you would interpolate between $\Phi(-1.41)=0.0793$ and $\Phi(-1.42)=0.0778$, giving $0.0793-0.49(0.0015)=0.07857$ — close to the exact value, with the difference due to table rounding.

4. 4

   State the answer.

   $P(X<40.851)\approx 0.0785\quad\text{(about }7.85\%\text{)}$

5. 5

   Check the result for plausibility. A $z$ of $-1.41$ lies just inside the classic $-1.645$ cut-off for the bottom $5\%$ and outside the $-1.28$ cut-off for the bottom $10\%$, so the tail area must fall between $5\%$ and $10\%$ — and $7.85\%$ does $\checkmark$. By symmetry, $P(X>69.149)$ has exactly the same value.