5.3.2 Normal Hypothesis Testing

How is a hypothesis test carried out with the normal distribution?

• The population parameter being tested will be the population mean, in a normally distributed random variable 
  • 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 
• 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 ≠ ..
• To carry out a hypothesis test with the normal distribution, the test statistic 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
• The normal distribution will be used to calculate the probability of the observed value of the test statistic taking the observed value or a more extreme value
• The hypothesis test can be carried out by
  • either calculating the probability of the test statistic taking the observed or a more extreme value (p – value) and comparing this with the significance level
  • or by finding the critical region and seeing whether the observed value of the test statistic lies within it
    • Finding the critical region can be more useful for considering more than one observed value or for further testing

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) find the distribution of the sample means, assuming H₀ is true, and use the inverse normal function on your calculator
• For a two-tailed test you will need to find both critical values, one at each end of the distribution

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 using the form

H₀ : μ = ...

H₁ : μ ... ...

Step 3.   Assuming the null hypothesis to be true, define the test statistic, usually 

Step 4.   Calculate either the critical value(s) or the p – value (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 p - value with the significance level

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

Worked Example