Math 143 C/E, Spring 2001
IPS Reading Questions
Chapter 7, Section 1 (pp. 502-507; up to “The one-sample t test”)



  1. This portion of Section 7.1 mirrors section 6.1 and has exactly the same purpose: to determine a confidence interval about a sample mean that has a certain likelihood of containing the true population mean. Why do we need another section devoted to this topic?















  2. As the authors say, there is a different t distribution for each different number of degrees of freedom. For a given sample, how would you determine the correct number of degrees of freedom to use?




  3. Just how similar or different are the t distributions from the standard normal distribution? To investigate this question I ask you to use several web applets for comparison. Click here to open up a browser window that runs the first applet. You will see two curves, one a student t curve and the other the standard normal curve. Using the slider you can alter the number of degrees of freedom for the student t and watch the curve change correspondingly. Start with the number of degrees of freedom at 1 and slowly move it up. While this particular applet may make the two curves appear the same more quickly than it should, you should at least get the right idea, that the student t curve becomes more and more like the standard normal curve as the number of degrees of freedom increases.

    After closing up the previous applet, click here and here to start up applets in two more windows. One applet draws the student t distribution, allowing you to set the degrees of freedom and move sliders around to highlight (in yellow) the region whose area you want to know. The other applet is similar, but uses the standard normal distribution. Set the degrees of freedom to 1 for the student t curve, and highlight the region between 0 and 1 (these values located along the horizontal axis). Compare the percentage of area between 0 and 1 for the two curves. Keep raising the number of degrees of freedom for the t curve until the amount of area is pretty much the same. What was the least number of degrees of freedom required? If you consider the area between 0 and 2 at this number of degrees of freedom, is it still the same?




  4. How does the calculation of margin of error change from the way we learned it in Chapter 6 when a t distribution is required.








  5. Just as with the confidence interval procedures we've learned that use the standard normal curve (Table A), those procedures that require the t(k) distribution require us to have a correspondence between levels of confidence C and critical (t*) values. We get these values from Table D. Example 7.1 indicates that the correct critical number for 7 degrees of freedom and 95% confidence is 2.365. Look at Table D, find the df = 7 row and read across until you reach the number 2.365. Notice that it is located in the probability = 0.025 column (100% - 95% leaves 5%, half on one end and half on the other end of the curve; you choose the column based on how much of the area you expect to be at just the extreme right end, which is 2.5%). To practice your use of Table D, fill in the critical numbers t* for the corresponding number of degrees of freedom and confidence levels:
    df C t*
    17 90%
    28 99.5%
    80 98%