Homework 2#

Problem 1: Hypothesis Testing#

Using the observations of peak flow data for the Sauk River, we are going to investigate whether a change in flood statistics occurred around 1977.

A. Descriptive Plots: Decide which of your plots from Homework 1 are relevant to the question of whether a change in flood statistics occurred in the Sauk River around 1977.

  • Include these plots from Homework 1 here.

  • Discuss what you can see visually in the graphs that would lead you to believe that a change has or has not occurred.

B. Two-sample test for a change in the mean: Test for statistical significance of the observed change in the mean annual peak flow around 1977.

  • Use a two sample test, and alpha = 0.05 (95% confidence) and the z-distribution to define the rejection region.

  • Discuss why using the z-distribution is appropriate here.

  • Use the two-sample test to compare the data from water years 1977-2023 to the data from water years 1929-1976, accounting for the different sample sizes and sample standard deviations appropriately (remember to use the “pooled standard deviation”).

  • For your null hypothesis, postulate that the difference between the two means = 0, and state the alternative hypothesis that the difference has changed (although you don’t know the direction of this change) and state the test statistic you’ll be using.

  • Can you reject the null hypothesis?

  • Calculate P after your test.

  • How does your estimate of P change if your null hypothesis is that the difference in the mean between the two data sets is equal to 15% of the pre-1977 sample mean? (In other words, test with a new null hypothesis: the mean of the second period is 1.15 times the mean of the first period.)

Problem 2: Test for a change in the variance or standard deviation:#

Lastly test for statistical significance of a change in the standard deviation. Even though it is not strictly true, assume that the sample data are derived from a normally distributed population. (While not required for this homework, we can follow up on the importance of this assumption by using Monte Carlo Tests as shown in Lab 2-3.)

Following lab 2-3, compare the results of using a single sample test vs. a two sample test. Using a single sample test (with rejection region based on the Chi Squared distribution), assume that the sample standard deviation that you calculated from the 1929-1976 data is close to the true population standard deviation that you are testing for a change from. For the two-sample test, assume both periods are independent samples.

Problem 3: Course Project Selection (CEWA 565)#

Upload this part of the assignment as a separate PDF or Word document to “Homework 2 Project Update” on Canvas.

Write a draft abstract for your project report (e.g. one paragraph of about 5-9 sentences). Describe your term project: what problem(s) will you be addressing, what data will you analyze, what question(s) will you answer, and what is your hypothesis? (You will add information about methods and results to your abstract later.) Be sure to include the name of your project partner in your submission (or specify that you will be doing the project alone).

If you have questions about the project, now is the time to talk to the instructor.

(the rubric for the final report is available for your reference here)