# 3.3.1 Normal Hypothesis Testing

## Normal Hypothesis Testing

#### What steps should I follow when carrying out a hypothesis test for the mean of a normal distribution?

• Following these steps will help when carrying out a hypothesis test for the mean of a normal distribution:
• Step 1.  Define the distribution of the population mean usually
• Step 2.  Write the null and alternative hypotheses clearly
• Step 3.   Assuming the null hypothesis to be true, define the statistic
• Step 4.   Calculate either the critical value(s) or the probability of the observed value for the test
• Step 5.   Compare the observed value of the test statistic with the critical value(s) or the probability with the significance level
• Or compare the z-value corresponding to the observed value with the z-value corresponding to the critical value
• Step 6.   Decide whether there is enough evidence to reject H0 or whether it has to be accepted
• Step 7.  Write a conclusion in context

#### How should I define the distribution of the population mean and the statistic?

• The population parameter being tested will be the population mean, μ  in a normally distributed random variable N (μ, σ2)

#### How should I define the hypotheses?

• A hypothesis test is used when the value of the assumed population mean is questioned
• The null hypothesis, H0 and alternative hypothesis, H1 will always be given in terms of µ
• Make sure you clearly define µ before writing the hypotheses, if it has not been defined in the question
• The null hypothesis will always be H0 : µ = ...
• The alternative hypothesis will depend on if it is a one-tailed or two-tailed test
• A one-tailed test would test to see if the value of  µ has either increased or decreased
• The alternative hypothesis, H1 will be H1 :  µ > ... or H1 :  µ < ...
• A two-tailed test would test to see if the value of µ has changed
• The alternative hypothesis, H1 will be H1 :  µ ≠ ..

#### How should I define the statistic?

• The population mean is tested by looking at the mean of a sample taken from the population
• The sample mean is denoted
• For a random variable  the distribution of the sample mean would be
• To carry out a hypothesis test with the normal distribution, the statistic used to carry out the test will be the sample mean,
• Remember that the variance of the sample mean distribution will be the variance of the population distribution divided by n
• the mean of the sample mean distribution will be the same as the mean of the population distribution

#### How should I carry out the test?

• The hypothesis test can be carried out by
• either calculating the probability of a value taking the observed or a more extreme value and comparing this with the significance level
• The normal distribution will be used to calculate the probability of a value of the random variable  taking the observed value or a more extreme value
• or by finding the critical region and seeing whether the observed value lies within it
• Finding the critical region can be more useful for considering more than one observed value or for further testing
• A third method is to compare the z-values of your observed value with the z-values at the boundaries of the critical region(s)
• Find the z-value for your sample mean using
• This is sometimes known as your test statistic
• Use the table of critical values to find the z-value for the significance level
• If the z-value for your test statistic is further away from 0 than the critical z-value then reject H0

#### How is the critical value found in a hypothesis test for the mean of a normal distribution?

• The critical value(s) will be the boundary of the critical region
• The probability of the observed value being within the critical region, given a true null hypothesis will be the same as the significance level
• For an %  significance level:
• In a one-tailed test the critical region will consist of  % in the tail that is being tested for
• In a two-tailed test the critical region will consist of  in each tail

• To find the critical value(s) use the standard normal distribution:
• Step 1.  Find the distribution of the sample means, assuming H0 is true
• Step 2.  Use the coding  to standardise to Z
• Step 3.  Use the table to find the z - value for which the probability of Z being equal to or more extreme than the value is equal to the significance level
• You can often find this in the table of the critical values
• Step 4.  Equate this value to your expression found in step 2
• Step 5.  Solve to find the corresponding value of
• If using this method for a two-tailed test be aware of the following:
• The symmetry of the normal distribution means that the z - values will have the same absolute value
• You can solve the equation for both the positive and negative z – value to find the two critical values
• Check that the two critical values are the same distance from the mean

#### Worked example

The time,  minutes, that it takes Amelea to complete a 1000-piece puzzle can be modelled using  .  Amelea gets prescribed a new pair of glasses and claims that the time it takes her to complete a 1000-piece puzzle has decreased.  Wearing her new glasses, Amelea completes 12 separate 1000-piece puzzles and calculates her mean time on these puzzles to be 201 minutes.  Use these 12 puzzles as a sample to test, at the 5% level of significance, whether there is evidence to support Amelea’s claim. You may assume the variance is unchanged.

#### Exam Tip

• Use a diagram to help, especially if looking for the critical value and comparing this with an observed value of a test statistic or if working with z-values.

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### Author:Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.