Normal Distribution - Calculations (Edexcel A-Level Mathematics): Revision Notes

Example: $X \sim N(30, 9)$

Find $P(X > 30)$.

Step 1: Sketch

• Sketch a normal distribution curve centered at $30$.
• The probability is represented by the area under the curve to the right of $30$.

Step 2: Use the calculator

• Access the calculator's distribution functions.

  

• Choose "Normal CD" for cumulative distribution.

• Input $\mu$ (mean), $\sigma$ (standard deviation), and bounds:

  

• Lower bound: $30$

• Upper bound: $999999$ (as an approximation of infinity)

• Input $\sigma$ (not $\sigma^2$).

Step 3: Find the result

• The probability, $P$, is calculated as $0.5$.

Example: $X \sim N(20, 2^2)$

Questions

a) $P(X > 22)$

b) $P(X < 18)$

c) $P(16 \leq X \leq 18)$

(a) Find $P(X > 22)$

• Use Normal CD with Lower: $22$, Upper: $999999$, and Standard Deviation: $2$.

  

• $P = 0.1586552539$ (b) Find $P(X < 18)$:

• Use Normal CD with Lower: $-999999$, Upper: $18$, and Standard Deviation: $2$.

  

• $P = 0.1586552539$ (c) Find $P(16 \leq X \leq 18)$:

• Use Normal CD with Lower: $16$, Upper: $18$, and Standard Deviation: $2$.

  

• $P = 0.1359051219$

Example: The heights of a large group of women are normally distributed with a mean of 165 cm and a standard deviation of $3.5$ $cm$. A woman is selected at random from this group.

Questions:

a) Find the probability that she is shorter than $160$ $cm$.

b) Steven is looking for a woman whose height is between $168$ $cm$ and $174$ $cm$ for a part in his next film. Find the proportion of women from this group who meet Steven's criteria.

c) A sample of $20$ women is taken from the group. Find the probability that at least $5$ of the women meet Steven's criteria.

a) Find the probability that she is shorter than $160$ $cm$.

• Use Normal CD with Lower: $-999999$, Upper: $160$, Mean: $165$, and Standard Deviation: $3.5$.

  

• $P = 0.07656372552$

• (Also, we could interpret this as meaning $7.66$% of women are shorter than $160$ $cm$)

b) Steven is looking for a woman whose height is between $168$ $cm$ and $174$ $cm$ for a part in his next film. Find the proportion of women from this group who meet Steven's criteria.

• Use Normal CD with Lower: $168$, Upper: $174$, Mean: $165$, and Standard Deviation: $3.5$.

  

• $P = 0.1906189738$

c) A sample of $20$ women is taken from the group. Find the probability that at least $5$ of the women meet Steven's criteria.

• $X \sim B(20, 0.1906)$
• $P(X \geq 5) = 1 - P(X \leq 4)$
• $P(X \leq 4) = 0.6703$
• $P(X \geq 5) = 1 - 0.6703 = 0.3296$