Finding the Parameters of a Normal Distribution (Cambridge International A-Level Mathematics)

Finding Sigma and Mu

How do I find the mean (μ) or the standard deviation (σ) if one of them is unknown?

• If the mean or standard deviation of the $X \text{tilde} N ( \mu , σ ² )$ is unknown then you will need to use the standard normal distribution

• You will need to use the formula

  • $z=\frac{x-μ}{σ}$ or its rearranged form $x=μ+σz$

• You will be given a probability for a specific value of $x(P(X<x)=p\mathrm{or}P(X>x)=p)$

• To find the unknown parameter:

• STEP 1: Sketch the normal curve

  • Label the known value and the mean

• STEP 2: Find the z-value for the given value of $x$

  • Use the table of the Normal Distribution to find the value of z such that $P(Z<z)=p$ or $P(Z>z)=p$

  • Make sure the direction of the inequality for $Z$ is consistent with $X$

  • The table gives the $z$-value to three decimal places to avoid rounding errors

    • Use the sketch to help you decide whether your z value is positive or negative
    • You should use the 3 decimal places throughout your calculations so that your final answer can be rounded to 3 significant figures

• STEP 3: Substitute the known values into $z=\frac{x-μ}{σ}$ or $x=μ+σz$

  • You will be given x and one of the parameters (μ or σ) in the question
  • You will have calculated z in STEP 2

• STEP 4: Solve the equation

How do I find the mean (μ) and the standard deviation (σ) if both of them are unknown?

• If both of them are unknown then you will be given two probabilities for two specific values of $x$

• The process is the same as above

  • You will now be able to calculate two z-values
  • You can form two equations (rearranging to the form $x=μ+σz$ is helpful)
  • You now have to solve the two equations simultaneously (you can use your calculator to do this)
  • Be careful not to mix up which z-value goes with which value of $x$

Worked Example It is known that the times, in minutes, taken by students at a school to eat their lunch can be modelled using a normal distribution with mean μ minutes and standard deviation σ minutes.

Given that 10% of students at the school take less than 12 minutes to eat their lunch and 5% of the students take more than 40 minutes to eat their lunch, find the mean and standard deviation of the time taken by the students at the school.

Answer:

Exam Hint

• These questions are normally given in context so make sure you identify the key words in the question. Check whether your z-values are positive or negative and be careful with signs when rearranging.