Hypothesis Testing

Hypothesis Testing

1. Basic Ideas of Hypothesis Testing

   • a hypothesis is a statement which may be true or may be false - it is not yet known
   • the null hypothesis is one where there is no change
   • the alternate hypothesis is one where there has been a change
   • the level of significance is the probability of the result where the null hypothesis would be rejected

2. Mean of a Normal Distribution with Known Variance

   • the test statistic is the mean and is used to decide between the two hypotheses
   • the critical value is the value at which the null hypothesis is rejected if it is larger or smaller then
   • the critical region is the region that contains all the values for when the null hypothesis is rejected
   • the acceptance region is the region that contains all the values for when the null hypothesis is accepted
   • the process of carrying out a hypothesis test is:
     • 1. declare the variable X - including a time or distance limit
     • 2. declare the null hypothesis and the alternate hypothesis
     • 3. declare the significance level
     • 4. carry out the test
     • 5. decide which hypothesis to accept and write a conclusion
   • a one-tailed test is when the alternative hypothesis is of the form u > or u <
   • a two-tailed test is when the alternative hypothesis is of the form u =/=
   • to test a hypothesis about the mean of a normal distribution with known variance, the standard normal distribution is used to find the critical region
     • if the value of the critical region = test statistic - mean / standard error, the null hypothesis is rejected

3. Mean of a Normal Distribution with Unknown Variance

   • to test a hypothesis about the mean of a normal distribution with unknown variance, the t-distribution is used to find the critical region
   • the variance is estimated is estimated using the s x s = (sum of x times x / n - 1) - (mean x mean)
   • if the value of t for the sample, where t = test statistic - mean / standard error, is in the critical region, then null hypothesis is rejected

4. Using Normal Approximation

   • if the distribution from which a random sample is taken is unknown, then provided the sample is sufficiently large (n >= 30) the z-statistic (variance known) or the t-statistic (variance unknown) can be used to test a hypothesis about the population mean

5. Type I and Type II Errors

   • a Type I error is to reject the null hypothesis when it is true - its size is equal to the significance level
   • a Type II error is to reject the alternative hypothesis when it is true