Edexcel A Level Maths: Statistics:复习笔记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

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