Hypothesis Testing for the Mean of a Normal Distribution (Cambridge (CIE) A Level Maths): Revision Note

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