STA240/ENV298.01 Homework 1

• Due Friday, August 31, 2001, 5pm. Submit homework to Sandra McBride's mailbox in the NSOE faculty mailroom (door to the right of the door to the NSOE reading room, box is in lower shelf on right).
• Review for Week 1:
  • Chapter 1 of Statistical Sleuth
  • All conceptual exercises in Statistical Sleuth (at end of chapter, not to turn in)
  • Review your errors on the diagnostic exam (available here), and do readings/problems to review skills.
• This homework is based on prerequisite material. Emphasis is given to the following sections of Moore and McCabe: 5.2, 6.1, 6.2, 6.3, 7.1.
• Homework 2 will be posted Thursday, August 30th and will be due the following week.
• Please note TA office hours.

1. Suppose the mean value of interpupillary distance (distance between eyes) for a species of the adult male gorilla is 65 mm, and the population standard deviation is known to be 5 mm.

   1. If the distribution of interpupillary distance is normal and a sample of n=25 males is selected, what is the probability that the sample average distance for these 25 will be between 64 and 67? At least 68 mm? What is the interquartile range (IQR) for the sample average of these data? If the sample size increased, how does the interquartile range for the sample average change?
   2. Suppose a sample of 100 adult males is obtained. In this case do not assume that the interpupillary distance is normally distributed. Can you repeat the question above for this sample size? If so, give your answer. If not, give your reasoning.

2. In "Effects of Roadside Conditions on Plants and Insects" (J. Appl. Ecology (1988):709-715) researchers reported the results of an experiment to evaluate the effect of nitrous oxide from automobile exhaust on roadside soil. The concentration of soluable nitrogen (in mg/g) was recorded at 20 roadside locations. We will consider data in the dataset noxroad.txt.
   1. Using Splus, give mean, median, standard deviation and interquartile range for this dataset. See Splus directions below

   2. Calculate by hand a 95% confidence interval for the mean concentration of soluable nitrogen. Show all steps. You should know how the standard deviation is calculated from the data, but for the purpose of this exercise, you can use the standard deviation printed by Splus. Confirm that your answer is identical to that generated by Splus.

   3. Give a 1-sentence interpretation of what the interval in (b) means.

   4. State any assumptions that are needed for inference about the above interval in (b) to be valid.

   5. Now assume that the standard deviation is known to be 0.15. Should you get a different answer for the confidence interval for the mean? If so, calculate it. If not, explain why.

   6. Two analysts perform separate hypothesis tests. (Assume unknown standard deviation as in (b).) Analyst 1 wants to know whether there is sufficient evidence to suggest that the mean differs from 1.69 mg/g. Analyst 2 evaluates the claim that the mean concentration is greater than 1.69 mg/g. For each analyst, write out the hypothesis test, giving null and alternative hypotheses, test statistics, and p-values. Are their conclusions the same or different? Why?

   7. Create 3 plots of the data: a boxplot, a histogram of counts and a frequency histograms. Label axes with units of measurement, and give titles to the graphs. You do not need to turn this in, but it will be assumed that you know how to do this computing task for the next homework.
      Splus directions: Save the dataset to your computer. Go to "File" and "Import Data" and "From File". Then select the directory that the file is in and select the format under "Files of type" by choosing "ASCII". Click on the column name, "V1", which is the default. Type in an appropriate variable name, like "concentration".
      Go to "Statistics" - "Data Summaries" - "Summary Statistics". Choose the variable name you would like to summarize, "concentration". Click on the tab called "Statistics" and check these boxes, "mean", "standard error", "confidence interval for mean", "1st quartile", "median", "3rd quartile".

      Additional directions for calculating a confidence interval for the mean in Splus and finding quantiles and percentiles of the standard normal and t distributions.

      To make the plots, go to "Graph" - "2D Plot" - "Histogram". Under "X column", select the variable of interest.

      • For the first histogram, go to the "Options" tab and chose "Output type" "Counts"

      • For the second, go to the "Options" tab and chose "Output type" "Freq". Make sure you understand what these two presentation methods say.

      • Add a title to your histogram by going to "Insert" "Text".

      For the boxplot, go to "Graph" - "2D Plot" - "Boxplot". Under "Y column", select the variable of interest.

      • Click on the y-axis label to add units of measurements.