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 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?H_1?:??μ > ...?or?H_1?:??μ < ...
  • A two-tailed test would?test to see if the?value of?μ?has?changed
    • The?alternative hypothesis,?H₁?will be H_1?:??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?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
• • • • 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
• 

• To find the critical value(s) use the standard normal distribution:
  • Step 1.? Find the distribution of the sample means, assuming H₀?is true
• 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