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

Can I use the standard normal distribution,?Z?, to perform a hypothesis test?

• You could use the standard normal distribution:
  • Find the z-value for your sample mean using
  • Find the critical value(s) for the Z distribution using the percentage points table
  • If the z-value is further away from 0 than the critical value then reject H0
• You could use the standard normal distribution as an alternative method for finding the critical value(s) is to 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 percentage points 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
  • 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

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
• Alternatively, if you have used the standard normal distribution method then in steps 4 and 5 you could compare the z – value corresponding to the observed value with the z – value corresponding to the critical value