3.3.1 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