Module 8

A Queen's University ecologist is studying a local population of red-winged blackbirds at the Queen's University Biological Station. She is interested in whether production of corticosterone, a common stress hormone in many animals, changes in the presence of a predator. To test this, she takes blood samples from 30 birds at two time points. The birds are left alone in an outdoor enclosure for 15 minutes before the first blood sample is taken. Following that, the birds are left in the same enclosure but a model predator (a plastic snake) is added to the enclosure to induce a stress response. The second blood sample is taken 15 minutes after the model predator is added. The following dataset contains levels of corticosterone in the blood samples of each bird taken at the two time periods. Use the data to answer the following questions. The ecologist intends to conduct a t-test to compare whether there is a difference in the mean corticosterone levels between treatments. Based on the above description, what is her null hypothesis? 1) A. Corticosterone levels will increase in the presence of a predator. B. Corticosterone levels will not change in the presence of a predator. C. Corticosterone levels will decrease in the absence of a predator. D. Corticosterone levels will not change in the absence of a predator. E. There is no null hypothesis for this test. 2) Given the null hypothesis and the study design, what type of test should she use? A. A one-tailed paired-sample t-test B. A two-tailed paired-sample t-test C. A two-tailed single-sample t-test D. A one-tailed two-sample t-test E. A two-tailed two-sample t-test

1 - B, 2 - B

A student advocacy group is interested in whether student performance on a second year philosophy exam is affected by which professor teaches the course. To test this, they randomly sample 50 exam grades from students who were taught by Professor E. Z. Going and another 50 exam grades from students who were taught by Professor Will Failue. 1) The student group intends to conduct a t-test to compare the exam grades. Based on the above description, what is the alternative hypothesis? A. There is no difference in the mean grades between Professors. B. Students taught by Prof. Will Failue have a lower mean grade. C. Students taught by Prof. E.Z. Going have a lower mean grade. D. There is a difference in the mean grades between Professors. E. Students taught by Prof. Will Failue have a higher mean grade. 2) Given the null hypothesis and the study design, what type of test should they use? A. A one-tailed paired-sample t-test B. A two-tailed paired-sample t-test C. A two-tailed single-sample t-test D. A one-tailed two-sample t-test E. A two-tailed two-sample t-test

1 - D, 2 - E

One difference and one similarity between "Hypothesis Testing" and "Estimation" for a population mean

1) Hypothesis testing leads to a yes/no (reject/fail to reject) decision. Parameter estimation produces a numeric value (e.g. estimate of the population mean), or a pair of values (.e.g confidence interval for the population mean), but does not result in an immediate yes/no decision. 2) Hypothesis testing answers the question "is a pre-determined population mean likely, based on what we saw in the sample?" Estimation of the mean provides an estimate of the population mean, working from the information in the sample. Similarities: 1) Both hypothesis testing and estimation are done using one sample from the population to make an inference about the population. 2) Both hypothesis testing and estimation rely on the sample being randomly selected from the population. 3) Both hypothesis testing and estimation make use of the sample mean and the standard error from the sample as part of their calculations. They just finish up with different calculations afterwards. 4) Both hypothesis testing and estimation involve the concept of the sampling distribution, and relating the probability of a 'rare event' in the tails of that sampling distribution to what was observed in the sample.

The mean annual consumption of beer in Ottawa is actually _less_ than the national mean. If the result of measuring a sample _does_ lead to the conclusion that the mean annual consumption of beer in Ottawa is below the national mean, this is a: A. Correct decision B. Type I error C. Type II error

A

The mean annual consumption of beer in Ottawa is actually _not less_ than the national mean. If the result of measuring a sample _does not_ lead to the conclusion that the mean annual consumption of beer in Ottawa is below the national mean, this is a: A. Correct decision B. Type I error C. Type II error

A

The mean body temperature for humans in fact differs from 37.1 degrees Celsius. If the result of measuring a sample _does lead_ to that conclusion, this is a A. Correct decision B. Type I error C. Type II error

A

The mean body temperature for humans is in fact 37.1 degrees Celsius. If the result of measuring a sample _does not_ lead to the rejection of the fact that the mean body temperature is 37.1 degrees Celsius, this is a A. Correct decision B. Type I error C. Type II error

A

The mean body temperature for humans is in fact 37.1 degrees Celsius. If the result of measuring a sample does not lead to the rejection of the fact that the mean body temperature is 37.1 degrees Celsius, this is a a) Correct decision b) Type I error c) Type II error

A

The mean cost to community hospitals per patient per day in Ontario actually _does not_ exceed the national mean. If the result of measuring a sample does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ontario exceeds the national mean, this is a A. Correct decision B. Type I error C. Type II error

A

The mean cost to community hospitals per patient per day in Ontario actually exceeds the national mean and the result of measuring a sample leads to the conclusion that the mean cost to community hospitals per patient per day in Ontario exceeds the national mean, this is a A. Correct decision B. Type I error C. Type II error

A

Which of the following is the correct way to interpret confidence intervals? Assume that the mean is 24.5 with 95% confidence intervals of +/- 3.5. a) If we repeatedly sample in the same way, 95% of the confidence intervals will include the true mean b) We can say with 95% certainty that a randomly sampled unit will be between 21 and 28 c)95% of the time, the sample mean will be between 21 and 28 d) 95% of the time, the true mean falls between 21 and 28 e) 95% of the time, the sample mean will be 24.5

A

The mean annual consumption of beer in Ottawa is actually _not less_ than the national mean. If the result of measuring a sample _does lead_ to the conclusion that the mean annual consumption of beer in Ottawa is below the national mean, this is a: A. Correct decision B. Type I error C. Type II error

B

The mean body temperature for humans is in fact 37.1 degrees Celsius. If the result of measuring a sample leads to the conclusion that the mean body temperature for humans differ from 37.1 degrees Celsius, this is a A. Correct decision B. Type I error C. Type II error

B

The mean cost to community hospitals per patient per day in Ontario actually _does not_ exceed the national mean. If the result of measuring a sample leads to the conclusion that the mean cost to community hospitals per patient per day in Ontario exceeds the national mean, this is a A. Correct decision B. Type I error C. Type II error

B

The mean annual consumption of beer in Ottawa is actually less than the national mean. If the result of measuring a sample _does not_ lead to the conclusion that the mean annual consumption of beer in Ottawa is below the national mean, this is a: A. Correct decision B. Type I error C. Type II error

C

The mean body temperature for humans in fact differs from 37.1 degrees Celsius. If the result of measuring a sample _fails to lead_ to that conclusion, this is a A. Correct decision B. Type I error C. Type II error

C

The mean cost to community hospitals per patient per day in Ontario actually exceeds the national mean. If the result of measuring a sample does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ontario exceeds the national mean, this is a A. Correct decision B. Type I error C. Type II error

C

Consider two samples taken from a population: Sample A has 20 observational units, and the sample mean is x¯ and std. dev s. Sample B has 200 observational units, and say it turns out to have the same sample mean x¯ and std. dev s as Sample A. (Say they are not identical, but just very close, and close enough for this argument to be treated as equal). a) Would having (almost) the same mean and standard deviation between the two samples with different sample sizes be reasonable, or wildly unlikely? b) Which sample will have the smaller standard error, and why? c) Which sample will have the narrower 95% confidence interval for the mean, and why? d) Why might you want to have a narrower confidence interval when doing statistical inference?

a) Since the samples come from the sample population, and the means and standard deviations of samples generally reflect those of the population they come from, it is not unreasonable that we would get two samples with different sizes to have similar means and standard deviations. b) The sample with 200 observational units will have the smaller standard error sx¯. The standard error (=standard deviation of the mean) = s/√n, the sample with the greater n (sample size) will have the smaller standard error. c) The sample with 200 observational units will have the narrower confidence interval. The confidence interval is x¯ ± t0.05(2),df × sx¯, and so with a smaller sx¯ for the larger sample size, you will also get a narrow confidence interval. (Note that the t0.05(2),df values will be slightly different between the two samples as well, but that effect is much smaller than the change in the sx¯ values due to the sample size increase. d) If you have a narrower confidence interval for a variable, then if your null hypothesis is not true, you are more likely to collect a sample that leads to the null hypothesis mean being outside the confidence interval, meaning that you would correctly reject the null hypothesis. With a wider confidence interval, the null hypothesis mean is in principle more likely to fall in the interval, and so you might fail to reject the null, even though it isn't true. We will be considering this issue of comparing the "reject/fail to reject the null" outcomes from the "null is true/false in reality" in more detail later on this course.