Statistical inferenceDraw conclusions from a set of data | Put a probability on whether a conclusion is correct ‘beyond reasonable doubt’ | The major question to answer is whether a difference between samples, or between a sample and a population, has occurred simply as a result of natural variation or because of a real difference between the two |
Two-tailed or one-tailedThe alternative hypothesis may be classified as two-tailed or one-tailed | Two-tailed test | - | is a two-sided alternative | - | we do the test with no preconceived notion that the true value of μ is either above or below the hypothesised value of μ 0 | - | the alternative hypothesis is written: H1: µ =/= µo | One-tailed test | - | one-sided alternative | - | do the test with a strong conviction that, if H0 is not true, it is clear that m is either grater than µ0 or less than µ0 | - | E.g. the alternative hypothesis is written as: H1: µ > µo |
| | Decision-making process steps1. | Collecting the data | 2. | Summarising the data | 3. | Setting up a hypothesis (i.e. a claim or theory), which is to be tested | 4. | Calculating the probability of obtaining a sample such as the one we have if the hypothesis is true | 5. | Either accepting or rejecting the hypothesis |
Significance levelAfter the appropriate hypotheses have been formulated, we must decide upon the significance level (or α -level) of the test | most common significance level used is 0.05, commonly written as α = 0.05 | A 5% significance level says in effect that an event has occurred that occurs less than 5% of the time is considered unusual |
One-sample z-testDeals with the case of a single sample being chosen from a population and the question of whether that particular sample might be consistent with the rest of the population | Construct a test statistic according to a particular formula | Information required in calculation | - | the size (n) of the sample | - | the mean of the sample | - | the standard deviation (s) of the sample | Other information of interest might include: | - | Does the population have a normal distribution? | - | Is the population’s standard deviation known? | - | Is the sample size (n) large? (25+) | There are different cases for the one-sample z-test statistic | Case I | the population has a normal distribution and | the population standard deviation, s, is known | Case II | the population has any distribution | the sample size, n, is large (i.e. at least 25), and | the value of population standard deviation is known | In both these cases we can use a z-test statistic formula (a) | Case III | the population has any distribution | the sample size, n, is large (i.e. at least 25), and | the value of population standard devation is unknown (however, since n is large, the value of population standard devation is approximated by the sample standard deviation, s) | In this case we can use a z-test statistic formula (b) |
| | Set up your HypothesisNull Hypothesis | Part of formulation of an hypothesis | Statement that nothing unusual has occurred | The notation is Ho | Alternative hypothesis | States that something unusual has occurred | The notation is H1 or HA | Together they may be written in the form: Ho: (statement) v. H1(alternative statement) |
Conclusion errorsTwo possible errors in making a conclusion about a null hypothesis | Type I errors occur when you reject H0 (i.e. conclude that it is false) when H0 is really true. | Type II errors occur when you accept H0 (i.e. conclude that it is true) when H0 is really false. |
z-test statistic formula (a)z-test statistic formula (b) |
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