# Hypothesis tests - T-tests, Z-tests, and percentage variations

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From the 14-10-13 Experimental Design and Analysis lecture.

## Resource summary

 Question Answer What is the first step of a hypothesis test? Write the null and alternative hypotheses. What is the difference between a one-sided alternative hypothesis and a two-sided alternative hypothesis? A one-sided alternative hypothesis is the opposite of the null hypothesis and only allows for one outcome, e.g. if the null hypothesis states that a mean is >5 the alternative may be that the mean is <5. A two-sided alternative hypothesis is also the opposite of the null hypothesis but gives a wider range of possibilities i.e. the null hypothesis could be that the true mean is 5, while the alternative hypothesis states that the mean is not 5 - in this case, the mean could be more OR less than 5, whereas in the one-sided alternative hypothesis, the mean could only be <5. How does one calculate standard deviation? 1) Subtract the mean from each individual value to find the deviations from the mean. 2) Square the deviations. 3) Total this value. 4) Divide the total of the squared deviations by the sample number minus one. 5) Square root the last value. How does one calculate standard error of the mean? Standard error = standard deviation / square root of the sample number Once you have null and alternative hypotheses, a mean, a standard deviation, and a standard error, what is the next step in a hypothesis test? Calculating the test value. Z or T = (sample mean - 'true mean')/standard error Both are calculated the same way, it is just the value that they are compared to that results in them differing. What is the rule for comparison of the test statistic value with the critical value from the New Cambridge Statistical Tables? If the test statistical value (T or Z) is greater than the critical value for a given significance level (5%, 1%, 0.1%), the null hypothesis can be rejected at that significance level. How does the test being one-tailed or two-tailed in its alternative hypothesis affect how the critical value is looked up in the normal tables? Example = for 5% significance level. If the test is two-tailed, look up the 2.5% value as the 5% overall is divided equally over the normal distribution. If the test is one-tailed, look up the 5% value as, though you are still using a normal distribution, there is only one way that the alternative hypothesis can go. What is important to include in a conclusion about a hypothesis test? What level of evidence there is (p = 0.05 - evidence; p = 0.01 - strong evidence; p = 0.001 - very strong evidence) and what it means for the null or alternative hypothesis - BASED ON THE DATA THE TEST WAS PERFORMED ON. You can never conclude that you've proven something, only that you've found evidence for it. How does one calculate the standard error of a percentage/proportion? The square root of: (('true percentage' x (100 - 'true percentage')) / sample number e.g. for a sample of 800 birds in which 25% SHOULD be blue ... square root of: ((25% x (100-25))/ 800 When would one use a t value rather than a Z value? Z values are applicable in hypothesis tests involving samples >30, whereas t values take into account the sample size using 'degrees of freedom' (sample size - 1) and so can be used for hypothesis tests involving samples <30. This is important because normal distribution cannot be assumed around a mean with small samples. If you had Minitab output for a hypothesis test, how would you check the significance of a test statistic (when manually you would look up critical values)? Check the P value generated. How would you write that you've found a t value for a sample size of 7 at the 5% significance level in notation form? t0.05,6 Where the numbers are subscripted, and 6 is the degrees of freedom (sample size - 1). How does one calculate the confidence interval around a percentage? How does this differ from the confidence interval around a mean? Multiply the Z value for the significance level from table 5 of the New Cambridge Statistics Tables by the standard error of the percentage e.g. 1.96 x 3.95. This differs from calculating the standard error of the mean in two ways: 1) the NCST value is derived from a table specific for percentages, and 2) the standard error is specific to the percentage value. What is the equation for calculating the test statistic (t or Z) when testing if a percentage if compatible with a claimed figure e.g. if a drug reduces symptoms in a certain percentage of a population? t/Z = (sample percentage mean - 'true' percentage mean) / standard error of the percentage What is a p value? A representation of how much risk of being wrong there is in rejecting a null hypothesis, e.g. if p<0.05 we could reject the null hypothesis with a 5% risk of being wrong (wrong = actually the null hypothesis is true), BASED ON THE EVIDENCE AVAILABLE. If a p value is >0.05, what can we do/not do? What can we conclude? We cannot reject the null hypothesis and so the data is not deemed to be 'significant' - there is no evidence to support the alternative hypothesis. If a p value is <0.05 (or 0.01, or 0.001) what can we do/not do? What can we conclude? We are able to reject the null hypothesis at the 5%, 1%, or 0.1% significant level, and, respectively, conclude that there is evidence, strong evidence, or very strong evidence, for the alternative hypothesis based on the data available.

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