# 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
• 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.