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Solutions 6 (1) In a statistical test about µ the null hypothesis was rejected. Based on this conclusion, which of the following statements are true? (a) A type I error was committed. FALSE. When the null hypothesis is rejected (and we accept the null), there are two possible outcomes. We have either (a) correctly rejected null - that is we have correctly proven the null wrong or (b) mistakenly rejected the null - that is said the null was wrong when it was in fact correct. The probability of wrongly rejecting the null when it is correct is called a type I error. (b) A type II error was committed. FALSE If we reject the null we cannot reject have committed a type II error. A type II error is committed if we don’t reject the null when in fact the alternative is true. (c) A type I error could have been committed. TRUE. As we mentioned in (a), if we reject the null we could have committed a type I error. We will never know! (d) A type II error could have been committed. FALSE. As we mentioned in (b), if we reject the null we can never have committed a type II error. (e) It is impossible to have committed both a type I and type II error. TRUE. (f) It’s impossible that neither a type I or type II error was committed. FALSE. A type I or type II error could have been committed. (g) Whether an error was committed or not is unknown, but if an error was made, then it was a type I error. TRUE. See part (a). (h) Whether an error was committed or not is unknown, but if an error was made, then it was a type II error. FALSE. See part (b). (2) True or False? (a) In a level α = 0.05 test of a hypothesis, increasing the sample size will not affect the level of the test. TRUE. The level of the test does not depend on the sample size. (b) In a level α = 0.05 test of a hypothesis, increasing the sample size will not affect the power of the test. FALSE. As we increase the sample size the power of the test increases - recall the examples we did in class. In other words, if H0 : µ = 1 and HA : µ > 1 and we want to calculate the chance of detecting the mean µ = 2 (this the alternative is true), as we increase the sample size is becomes easier to detect that the population mean is 2. (c) The sample size n plays an important role in testing a hypothesis because it measures the amount of data (and hence information) upon which we base a decision. If the data is quite variable and n is small, it is unlikely that we will reject the null when when the null is false. TRUE. If the sample size is small and the data is highly variable (large standard √ deviation), then the standard error is large (recall the formula σ/ n, where σ is the variability of the data and n the sample size). A large standard error means there will a lot of variation in the sample mean. This makes it harder to detect what the true population mean is (a shielding effect), and thus harder to reject the null when the mean is different to what the mean null. (d) Supose we are testing the following hypothesis about the population mean µ, H0 : µ ≤ 4 versus HA : µ > 4. If the sample size is very large and the variable is not highly variable (hence small variance), it is highly likely we will reject the null hypothesis even when the true value of µ is only trivially larger than 4. TRUE. As we mentioned in the class, this is an example of impractical significance. It is rare in practice that the mean will ever equal exactly 4. For very large samples, the standard error of the sample mean will very small. Thus the estimate of the mean is likely to be close to the population mean. This means for large samples we are highly likely to reject the null even if the mean is µ = 4.0001 (trivially greater than 4.) (3) Import the calf weight data (it can be found by the side of this homework) into Statcrunch. It is believed that the weight of a newborn animal (in general) drops immediately after they are born and only after a few weeks does the weight get back to the birth weight and above. We want to investigate whether the calf weight data suggests this to be true. Look at the data in Statcrunchs. The column with Wt 0 contains all the weights at birth. Wt 0.5 contains the weights at week 0.5, Wt 1 the weights at week 1 etc. (a) Explain why there is matching for the weights of the calves at various weeks. We have the weight of calves over the week. Since the same calves are observed over the weeks, it is clear that there is a matching of the data sets over the weeks. Thus to watch the change in calf weights over the weeks one must consider the differences in weights. (b) For each of the following situations first write the correct hypothesis and then do the following tests at the 5% level (the Statcrunch instructions can be found on my slides). (i) Do a matched paired t-test to see if there is evidence to support the view that the weight has dropped between week 0 and week 1. Figure 1: Comparing between birth and Week 1 To see if there is a loss in weight as calves go from birth to week 1 we are testing H0 : µweek 0 − µweek 1 = 0 against HA : µweek 0 − µweek 1 > 0. The p-value is less than 0.1%, based on this, there is evidence to suggest that there is a loss in mean weight. (ii) Do a matched paired t-test to see if there is evidence to support the view that the weight has dropped between week 0 and week 2. Figure 2: Comparing between brith and week 2 To see if there is a loss in weight as calves go from birth to week 2 we are testing H0 : µweek 0 − µweek 2 = 0 against HA : µweek 0 − µweek 2 > 0. The p-value is less than 0.1%, based on this, there is evidence to suggest that there is a loss in mean weight. (iii) Do a matched paired t-test to see if there is evidence to support the view that the weight has dropped between week 0 and week 3. Figure 3: Comparing between birth and week 3 (one-sided test, for a decrease in weights) To see if there is a loss in weight as calves go from birth to week 3 we are testing H0 : µweek 0 − µweek 3 = 0 against HA : µweek 0 − µweek 3 > 0. The p-value is 72% based on this, there is evidence to suggest that there is a loss in mean weight. Looking at the numbers, it is clear that the p-value would be larger than 50%, since the difference is negative. There can’t be any evidence for a the mean differences to be positive when in fact the difference in the sample means is negative. We re-do the test, but this time looking for any difference (not just an increase or a decrease). This means testing H0 : µweek 0 −µweek 3 = 0 against HA : µweek 0 − µweek 3 6= 0. In this case the p-value is 54.8% and we cannot reject the null. This means there is no evidence to suggest that the mean weight of the calf is different to the mean birth weight. Figure 4: Comparing difference between birth weight and weight at Week 3 (looking for a difference). (iv) Do a matched paired t-test to see if there is evidence to support the view that the weight has increased between week 0 and week 3. This means testing H0 : µweek 0 − µweek 4 = 0 against H0 : µweek 0 − µweek 3 < 0. The p-value can be obtained directly by re-running Statcrunch, but can also be deduced from either of the outputs in part (iii). From these outputs (noting that now the alternative is pointing in the opposite direction), we can deduce that the p-value %). Thus there is no is 100-72.58 = 37.56% (or equivalently 54.82 2 evidence to reject the null. (v) Do a matched paired t-test to see if there is evidence to support the view that the weight has dropped between week 0 and week 4. Here we are testing H0 : µweek 0 − µweek 4 = 0 against HA : µweek 0 − µweek 4 > 0. From Figure 5 it is clear that the sample mean difference is negative (-8.3) and are testing for a positive difference in the population means, so there is no evidence to support the alternative. In fact we can also deduce that p-value for this hypothesis is greater than 99.99% (since it the t-stat - which is the real measure of distance is huge and the p-value in the opposite direction is less than 0.01%). This p-value can be obtained directly from the Statcrunch, but we have deduced the p-value using Figure 5, which gives the hypothesis in the opposite direction. (v) Do a matched paired t-test see if there is evidence to support the view that the weight has increased between week 0 and week 4. Now we want to see if there is a weight gain at week 4. This means testing the hypothesis. H0 : µweek 0 − µweek 4 = 0 against H0 : µweek 0 − µweek 4 < 0. We see from Figure 5 that the p-value is less than 0.1%, thus at the 5% level there is evidence to suggest that at week 4 the mean weight has increase from birth. Figure 5: (c) Based on your results in (b) summarise the mean behaviour of calf weights from week 0 to week 4. Write this as a mini report (say about 5 lines) summarising how you came to your conclusions. From the tests that we have done above, the data suggests that compared to their birth weight there is a drop in the weight of calves at week 1 and week 2, and from week 4 onwards the weight has returned and there is an increase from birth weight. What happens at week 3 is a little ambigious, we could not reject the null for both the one-sided and two-sided tests, this means there is no evidence to suggest the mean weight at week 3 is any different to the mean birth weight. We know that we cannot reject a null, but I would hazard a non-statistical guess that at week 3 the weight is back to the birth weight.