Geography 360:

Problem Set Six

1. a) In order to conduct a one sample difference of means test, we will return to the data on 5-year-old child's heights [113.7, 104.6, 108.6, 105.9, 110.0, 106.7, 108.5, 107.7, 114.3, 116.7, 103.5, 96.1, 110.8, 97.2, 109.6, 110.5, 105.9, 106.2]. Previously, we used this data to calculate probabilities (assuming a normal distribution) and confidence intervals. In this case, we will treat the data as a sample of five-year-olds who have consumed diets rich in camel's milk since birth. If the average height for all five-year-olds is 104.9 cm, construct your null and alternative hypotheses to determine if the five-year-olds in the sample were significantly taller than the population at large. Using the p method, calculate the t test for single sample difference of means (remember, you must calculate the mean and standard deviation the sample and then the do a t-test - sample size less than thirty at n-1 degrees of freedom) and state the significance of the p-value (remember that this is equal to the probability of being wrong when rejecting the null hypothesis, that is of a Type I error).

b) If the average height in the above test is for all five-year-olds and the sample is only of male children, how would the p-value change when compared to male children with a population average height of 105.5? How would this change your interpretation of the data?

2. a) The two sample difference of means test can be used to compare samples taken from a similarly defined population to establish the significance of change in a particular variable (in other words, is the population in year B different from year A relative to the variable studied). Returning to the table on rare animal sightings in Wisconsin used for the probability distributions, it is possible to compare the sightings in one year with those recorded several years later. If we assume again that sightings provide a relatively good estimate of population size, we can make a statistical statement about the growth of the population for a particular species. In this case, compare the sightings of gray (timber) wolves in 2000 (pg. 99 in report at this link) with those in 2002 (pg. 132-133 in report at this link). How confident can you be that the wolf population has grown during the two-year period. First state your null and alternative hypotheses. Again, using the p method, calculate the confidence level associated with rejecting the null hypothesis. Are you surprised with the outcome?

b) Given that the wolf population is related to available habitat in each county, it is possible to argue that the population is not normally distributed across counties. In other words, the number of sightings is not independent of the location in which sightings were recorded. Because this challenges the assumptions associated with the two sample difference of means test, recalculate the p-value using the Wilcoxian two sample mean rank test. How does your answer compare to that calculated with the t-test? Does it change your interpretation? If not/so, can you predict how the values in the second sample would have to change in order to alter your interpretation?