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
The time, minutes, that it takes Amelea to complete a 1000-piece puzzle can be modelled using . Amelea gets prescribed a new pair of glasses and claims that the time it takes her to complete a 1000-piece puzzle has decreased. Wearing her new glasses, Amelea completes 12 separate 1000-piece puzzle and calculates her mean time on these puzzles to be 201 minutes. Use these 12 puzzles as a sample to test, at the 5% level of significance, whether there is evidence to support Amelea’s claim. You may assume the variance is unchanged.
- Use a diagram to help, especially if looking for the critical value and comparing this with an observed value of a test statistic