Edexcel A Level Maths: Statistics:复习笔记4.3.3 Standard Normal Distribution
Standard Normal Distribution
What is the standard normal distribution?
- The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1
- It is denoted by Z
-
Why is the standard normal distribution important?
- Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch
- Therefore we have the relationship:
- Where
and 
- Probabilities are related by:
- This will be useful when the mean or variance is unknown
- If a value of x is less than the mean then the z-value will be negative
- Some mathematicians use the function
to represent 
The table of percentage points of the normal distribution
- In your formula booklet you have the table of percentage points which provides information about specific values of the standard normal distribution that correspond to commonly used probabilities
- You are given the value of to 4 decimal places when p is:
- 0.5, 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.025, 0.01, 0.005, 0.001, 0.005
- These values of z can be found using the "Inverse Normal Distribution" function on your calculator
- If you are happy using your calculator then you can simply ignore this table
- They are simply listed in your formula booklet as they are commonly used when:
- Finding an unknown mean and/or variance for a normal distribution
- Performing a hypothesis test on the mean of a normal distribution
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
is unknown then you will need to use the standard normal distribution - You will need to use the formula
or its rearranged form 
- You will be given a probability for a specific value of
or 
- 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 Inverse Normal Distribution to find the value of z such that
or
- Make sure the direction of the inequality for Z is consistent with X
- Try to use lots of decimal places for the z-value to avoid rounding errors
- You should use at least one extra decimal place within your working than your intended degree of accuracy for your answer
- Use the Inverse Normal Distribution to find the value of z such that
- STEP 3: Substitute the known values into
or 
- 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 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
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 standard deviation 4 minutes.
Given that 10% of students at the school take less than 12 minutes to eat their lunch, find the mean time taken by the students at the school.
Exam Tip
- 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.
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