STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...

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Week 7 H/Y TESTING HYPOTHESES ABOUT POPULATION MEANS i. Hypotheses about a population mean, µ, using one random sample of observations, without value for σ . Without a known value for the population standard deviation, σ, the hypothesis is tested using the standard deviation, s, of the random sample. For a sample size n, the sample estimate s has (n-1) degrees of freedom. The test statistic is no longer a z-statistic, but a t-statistic. Go back to the examples in part (i) on pages 10-11, and in the description suppose that the given value of σ is the sample estimate, s. Now answer the questions Q1 to Q12 using these values of s, and the appropriate tdistribution. ii. One random sample of items, two variables observed on each item, and we are interested in the difference between the means of the variables. The two variables are said to be ‘paired’ or ‘matched’ because they are observed on the same item. PAIRED t-TEST: QUESTION 1. (from Stat171 lecture overheads, prepared by Stephen Brown) In an experiment using 15 hypertensive patients, the effect of the drug Captopril on their blood pressure is assessed by recording their blood pressure, giving them Captopril and then measuring their blood pressure again two hours later. Does the drug have any effect on blood pressure? 10 Patient Blood pressure Blood pressure DIFFERENCE BEFORE drug AFTER drug BEFORE - AFTER 1 130 125 5 2 122 121 1 3 124 121 3 4 104 106 -2 5 112 101 11 6 101 85 16 7 121 98 23 8 124 105 19 9 115 103 12 10 102 98 4 11 98 90 8 12 119 98 21 13 106 110 -4 14 107 103 4 15 100 82 18 Notice that • there is only one random sample of patients • there are two observations on each patient and we are interested in the difference between the two • the change is expressed as the drop in blood pressure. Nan Carter: workshop notes prepared for Numeracy Centre Macquarie University.
We now have a single sample of differences, and the researcher is expecting that there will be a drop in blood pressure after taking the Captopril. The Statistical hypothesis is that for population of treated patients the mean drop is zero. So, H0 : μd = 0 Assumptions: we only assume that the differences are Normally distributed, and that the differences are independent. COMMENTS • To do a paired t-test, the experimenter must plan ahead to collect the data in the appropriate way. • As students in Stat170, you are expected to recognise paired data from the information given in the question. • Remember that you will be interested in the difference between the two observations, that is, the null hypothesis will be Ho: μd = 0. • The pairing is not always as obvious as the ‘before’ and ‘after’ example above; see for example, question 2, where we have observations on pairs of dogs, each pair taken from the same litter (dogs from the same litter have the same parents, and hence should be more alike genetically than dogs from different litters). Here, the random sample is of litters of dogs arriving at the pound. • See also question 2 in the 1998 mid-year examination, where you have a random sample of women and information on their incomes and the incomes of their partners: the question relates to the difference between a woman’s and her partner’s income. • ***** ASSUMPTIONS: We need make absolutely no assumptions about the original observations. If they are not normally distributed, it does not matter, as long as the differences are. Plot the differences to check that the assumption is not seriously violated. QUESTION 2. (from Stat171 tutorial exercises) In order to investigate whether the time of neutering of male dogs had any effect on their rate of growth, Crenshaw and Carter (1995) neutered dogs at 2 months and 7 months of age and measured their weight at 18 months of age. As litters of any breed became available at a pound, two male dogs were randomly selected from each litter. One was neuterd at 2 months, the other was neutered at 7 months. All the dogs were kept under identical conditions during the 18 months of the trial. The weights (in kg) of the dogs at 18 months were: Litter 1 2 3 4 5 6 7 8 9 10 2 months 19.0 23.2 16.1 30.3 27.1 22.3 18.1 27.8 34.2 17.3 7 months 18.9 23.5 16.7 30.3 27.8 22.4 18.2 28.1 34.1 17.7 a) Is there any evidence that early neutering affects the rate of growth of the dogs? b) Comment on why you think the experimenter chose this particular design in setting up the experiment. Do you think the right choice of design was made? 11 Nan Carter: workshop notes prepared for Numeracy Centre Macquarie University.
Page 1 and 2: STAT170 Workshop Notes prepared by
Page 3 and 4: Week 4 H/Y. Exercises for STAT170 w
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Page 7 and 8: EXAMPLES ON WORKING WITH PROPORTION
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Page 17 and 18: EXERCISE 11: In Q5 use a hypothesis
Page 19 and 20: PAIRED t-TEST: QUESTION 1. (from St
Page 21 and 22: QUESTION 4. “You can taste the lu
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Page 25 and 26: a) Draw the scatterplot of the data
Page 27 and 28: Residual 20 15 10 5 0 -5 -10 -15 -2
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Page 31 and 32: Relation: Monthly sales volume outc
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STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...

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