Hypothesis Testing for the Population Mean of a Normal Distribution (AQA A Level Maths: Statistics): Revision Note

Normal hypothesis testing

How do I do a hypothesis test with a normal distribution?

• The population parameter being tested will be the population mean, in a normally distributed random variable $N(\mu ,{\sigma }^2)$

  • The population mean is tested by looking at the mean of a sample taken from the population

  • The sample mean is denoted $\overline{x}$

  • For a random variable $X~N(\mu ,{\sigma }^2)$ the distribution of the sample mean would be $\overline{X}~N(\mu ,\frac{{\sigma }^2}{n})$

• 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, $\overline{\mathit{X}}$

  • 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 do I find the critical value?

• 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  $\alpha$%  significance level:

  • In a one-tailed test the critical region will consist of  $\alpha$% in the tail that is being tested for

  • In a two-tailed test the critical region will consist of $\frac{\alpha }{2}\%$ 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

How do I lay out the hypothesis test?

• 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 $X~N(\mu ,{\sigma }^2)$

• 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 $\overline{X}~N(\mu ,\frac{{\sigma }^2}{n})$

• 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