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?

    While the information in Section 6.1 is accurate, carrying out the confidence interval procedure that is described there requires that you know the standard deviation of the population. Now the whole setting assumes that you don't know (but want to know) something about the population mean. If you don't know the population mean, isn't it even less likely for you to know its standard deviation? While knowledge of the population standard deviation is probably impossible, you can calculate an approximation to it, namely s, the standard deviation for the sample. Using it (instead of s) you can also compute an approximate spread (referred to as the standard error of the sample mean; see p. 504) for the sampling distribution of the mean. Nevertheless, use of standard error in place of s introduces more complications than you might expect, the most important of which is that the sampling distribution is no longer a normal distribution. Rather, it is a t distribution.

  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?

    The number of degrees of freedom is one less than the sample size. For samples of size 20, you would use the t distribution with df = 19 (i.e., 19 degrees of freedom).

  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?

    The value of 34.1% that the standard normal curve gives isn't seen under the student t curve until you reach 144 degrees of freedom. Even at this level, the areas under the two curves between 0 and 2 do not match exactly.

  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.

    We learned in Chapter 6 (when we assumed that s was known) that the margin of error for population means is m = z* s. When a t distribution is required (that is, when s is not known), the critical z* value is replaced with the critical t* value and the standard deviation for the population s is replaced with that (s) for the sample.

  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%

    Answers: 1.740, 3.047, 2.374



File translated from TEX by TTH, version 2.87.
On 8 Mar 2001, 14:52.