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Edexcel A Level Maths: Statistics:復習筆記4.3.3 Standard Normal Distribution

Category: A-level課程, 國際課程, 教材筆記, 福利干貨 Date: 2022年11月7日上午10:08

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
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.

轉載自savemyexams