To Lecture Notes

IT 223 -- 3/9/11

Review Questions

1. What are the differences between a t-test and a z-test.

   Ans: A z-test must have n >= 30 and can be used to test null hypotheses for mu or for p. The data need not be normal. A t-test can be used when t < 30, but then the data must be close to normal. A t-test can be used to test null typotheses for mu, but not for p.

2. A dataset contains two values x₁ = 1500 and x₂ = 1510. The dataset is generated by the ideal measurement model. Test the null hypothesis that μ = 0 using a 99% confidence interval. Ans
   1. H0: mu = 0, H1: mu ~= 0

   2. xbar = 1505, SD+ = sqrt(50), SEave = SD+ / sqrt(2) = sqrt(50) / sqrt(2) = sqrt(25) = 5. t = (xbar - mu) / SEave = (1500 - 0) / 5 = 300.

   3. Degrees of freedom: n - 1 = 2 - 1 = 1. A 99% confidence interval from the t-table is [-63.66, 63.66].

   4. t = 300 ∉ [-63.66, 63.66], so reject the null hypothesis.

   5. SPSS gives the p-value as 0.002, which also means that H0 is rejected with 99% confidence interval.

3. In 1998, as an adverting campaign, a cookie company claimed that every 18-ounce bag contained an an average of 1000 chocolate chips. Students at the Air Force Academy in Colorado Springs bought some randomly selected bags of cookies and counted the chocolate chips. Some of their data are recorded on the ChocolateChips dataset of the Excel file.
   1. Form the normal plot of the chocolate chip counts. Are the counts normally distributed?

   2. Perform a t-test, using a 99% confidence interval, of the null hypothesis that the average chocolate chip count is 1000 per bag.

4. Human computer interaction researchers are measuring the time required to move a mouse cursor to a target on the screen. The positions of the cursor and the target are randomly initialized. Two types of mouse are used (1) a traditional mouse and (2) a touch sensitive mouse with gaze positioning using a commercial eye tracker. The dataset mouse-experiment.xls contains four columns t1, t2, t3, t4, with these meanings:

   Variable	Meaning
   t1	Time to position traditional mouse on blank background
   t2	Time to position touch sensitive mouse with gaze control on blank background
   t3	Time to position traditional mouse on complex background
   t4	Time to position touch sensitive mouse with gaze control on complex background

   1. Test the null hypothesis that, on a blank background, there is no difference between the time to move the mouse cursor to the target using a traditional mouse vs. using a touch sensitive mouse with gaze positioning.

   2. Test the null hypothesis that, on a complex background, there is no difference between the time to move the mouse cursor to the target using a traditional mouse vs. using a touch sensitive mouse with gaze positioning.

Degrees of Freedom

• Degrees of Freedom (df) is a technical term that arises when using the t-test.

• The degrees of freedom for a t-test is related to the n - 1 that is used in the denominator of SD+.

• The degrees of freedom are the number of observations minus the number of parameters that must be estimated when estimating the variance for a statistical model.

• For example, when estimating the variance of the errors for the ideal measurement model
  σ² = [(x₁ - x)² + ... + (x_n - x)²] / (n - 1),

  we must estimate the parameter μ with x, so we loose one degree of freedom from the n degrees of freedom gained by the n observersions.

• When estimating the variance for a regression model, we have n - 2 degrees of freedom: +n for the n observations; -2 for estimating the two parameters a and b in y = ax + b.

More about Hypothesis Tests

• Was the result important?
  Just because the result of a statistical test is significant does not mean that it was important.

  Suppose that an aspirin manufacturer claims that its product causes a headache to go away significantly faster than a headache when the product is not used.

  Looking at the data reveals that when the product is used, headaches go away in 6 minutes, on the average. When the product is not used headaches go away in 7 minutes, on the average. Even if this result is significant, is it important.

  Many tests of hypothesis are significant if n is large enough; this does not always prove importance.

• Data Snooping
  If you do a large number of statistical tests, about 5 out of 100 will be significant, just by chance, even if H₀ is true.

  Example: the Crestwood cancer cluster.

• One-tailed Tests
  Many statistics testbooks discuss one-tailed tests, with a null hypothesis of the form H₀: μ > a.

  While one-tailed tests are useful in many cases, in other cases they are merely thinly veiled attempts to get a significant result when the two-tailed test was not significant.

  This is because the p-value for a one-tailed z- or t-test is half of the value of the corresponding two-tailed test.

  • We are only considering two-tailed tests in this course.

t-tests with SPSS

• To perform a one-sample t-test with SPSS:
  Analyze >> Compare Means >> One-Sample T Test

• To perform a paired-sample t-test with SPSS:
  Analyze >> Compare Means >> Paired-Samples T Test

Review for Final Exam

• See the review materials on the Exam Info Page.