Mathematics 376 – Probability and Statistics II Lab Project 3 – Hypothesis Testing April 5 and 7, 2004

Background

We have now discussed various tests of hypotheses for means, variances, etc. of normal distributions. In this lab we will work through several realistic examples to illustrate the thinking process statisticians would use to select appropriate tests, and interpret the results.

New Version of Maple Package

Start by downloading the latest version of the Maple statistics package from the course homepage (version of March 16). There is a new graphical routine in this iteration that produces what are called “box-and-whisker” plots for lists of numerical data that you will want to use at several points in this lab. The idea is the following. To get a rough visual picture of the distribution of a data set, as an alternative to the relative frequency histogram, we can use a box-and-whisker plot. The idea of this graphical display is to show the locations of the minimum, the 25th percentile, the median or 50th percentile, the 75th percentile and the maximum of the data values by drawing a box with vertical bars at the 25th, 50th, 75th percentile values, together with two thinner “whiskers” extending out to the minimum and maximum. The new procedure in the Maple package is designed to draw one of these plots for any collection of lists, “stacked vertically,” in one graphical display so that we can visually compare data sets in a rough way. Check the on-line documentation for more information and a usage example.

Lab Questions

A) An office furniture manufacturer has developed a new glue application process for assembling tables. To compare the two processes, random samples with n = 30 were selected from inventories produced with the old process and with the new process. Each table was subjected to “destructive testing” in which the force (in pounds) needed to break the glue in the table was measured. Let X be the force in pounds needed to break one of the new tables, and Y be the force in pounds needed to break one of the old tables. The goal is to determine whether the new gluing process has significantly increased the strength of the tables. The data collected was as follows:

X : 1250 1210 990 1310 1320 1200 1290 1360 1200 1150 1120 1360 1310 1110 1320 980 950 1430 1100 1080 960 1050 1310 1240 1420 1170 1470 1060 1230 1300

and

Y : 1180 1360 1310 1190 920 1060 1440 1010 1000 950 1310 980 1310 1030 960 800 1280 1080 900 1030 930 1050 1010 1310 940 860 1450 1070 840 1100

1) Construct box-and-whisker plots for these data sets (plotted together) and make an informal conjecture about whether or not the new process (the X data) has increased the strength of the tables, compared with the Y data.

2) Describe an appropriate test of the null hypothesis H_{0} : μ_{X} = μ_{Y} versus H_{a} : μ_{X} > μ_{Y}. Say what your assumptions about the data are, what your test statistic is, what the rejection region will be, and so forth.

3) Carry out your test at the α = .05 level of significance. Give a clear and concise statement of the conclusion you draw from your test.

4) What is the attained significance level of your test (the p-value)? Use the appropriate CDF function in the Maple package to determine an accurate estimate of p, not just a range of possible values. What does this say?

B) Let X be the weight in grams of a Low-Fat Strawberry Kudo bar and let Y be the weight in grams of a Low-Fat Blueberry Kudo bar. Assume that the distributions of X, Y are normal: N(μ_{X}, σ_{X}^{2}) and N(μ_{Y} , σ_{Y}^{2} ) respectively. A random sample of n_{X} = 9 X weights were taken:

21.7, 21.0, 21.2, 20.7, 20.4, 21.9, 20.2, 21.6, 20.6

Similarly, a random sample of n_{Y} = 13 observations of Y were made:

21.5, 20.5, 20.3, 21.6, 21.7, 21.3, 23.0.21.3, 18.9, 20.0, 20.4, 20.8, 20.3

Random variations in the manufacturing processes mean that the weights of the bars are not always the same, and appear to vary depending on the type of bar as well. Is there a statistically demonstrable difference in the weight distributions of two types of bars, though? Here is one possible procedure for testing for equality of two normal distributions in the small sample case:

• First, test for equality of variances.

• If there is no demonstrable difference in the variances, test for equality of the means using the t-test for equality of means that we discussed in class (using the pooled estimator S_{p} for the common variance).

• If there is a demonstrable difference in the variances, the basic t-test is not all that reliable in many cases. With small sample sizes, most experienced statisticians would use a different approximating distribution due to Welch: Test for equality of means using a t-distribution with r degrees of freedom where

(1)  r = [\frac{(S_{X}^{2}/n_{X} + S_{Y}^{2} /n_{Y})^{2}}{(S_{X}^{2}/n_{X})^{2}/(n_{X} − 1) + (S_{Y}^{2} /n_{Y})^{2}/(n_{Y} − 1)}]

(where [z] = greatest integer less than or equal to z).

In this problem, you will follow this procedure to test this data:

1) Test the null hypothesis H_{0} : σ_{X}^{2} = σ_{Y}^{2} against the alternative hypothesis H_{a} : σ_{X}^{2} ≠ σ_{Y}^{2} . You may select the significance level α. Interpret your results.

2) Test the null hypothesis H_{0} : μ_{X} = μ_{Y} against the alternative hypothesis H_{a} : μ_{X}^{2} ≠ μ_{Y}^{2} . Select the test statistic, etc. based on the results of your test from part 1 and explain your choice. Also clearly state the conclusion you draw from the test.

3) Construct box-and-whisker plots of the two data sets and reconcile with your results in parts 1 and 2.

Assignment

Group write-ups due at the end of the class on Wednesday, April 7.