Math 143 C/E, Spring 2001
IPS Reading Questions
Chapter 5, Section 1 (pp. 372-385)



  1. What is a population distribution of a variable? How does it differ from a sampling distribution? Does the distinction between a population that does exist (say, women between the ages of 18 and 24) and one that doesn't (all rods that would be produced by a certain manufacturing process if it continued forever) bother you?

    A sampling distribution for a statistic is, in fact, a probability distribution, an idealization, a mathematical model. When you take a sample of size n, measure each unit in the sample for the desired variable, compute the sample statistic, and then repeat the process over again many times, the plot of the distribution of resulting statistics should begin to look like the sampling distribution (and look more and more like the more times you repeat the exercise). (See the sampling distributions applet of the 2/15/2001 lecture.)

    The population distribution is what you would have if you took a census and plotted its distribution. (Such pictures aren't very interesting for categorical variables.) If you were to do the sampling exercise described in the last paragraph, but each time your sample was of size n = 1 (so that the statistic you plot for each sample is simply the one individual value), then the emerging distribution would begin to look more and more like the population distribution.

  2. Compare Examples 5.2(b) and 5.3. The authors say emphatically that 5.2(b) is not a binomial setting, while 5.3 is “not quite a binomial setting.” Don't these scenarios seem similar? How do they justify letting it go in 5.3, but not in 5.2(b)? Do they indicate how you would decide whether a certain binomial setting is close enough to binomial to let it go?

    The two scenarios are very similar - choosing a member of the population for the sample and removing that member (so as not to be chosen twice for the same sample) changes the probabilities for further choices. The authors say that as the ratio of population size to sample size increases, it becomes more and more valid to overlook this lack of independence in the choosing of members in a sample. The rule of thumb they give is a ratio of 10:1. The ratio in 5.2(b) is 52 (# of cards in deck = population) to 10 (# of cards in sample), which is too small to overlook the lack of independence; they assume in 5.3 that the ratio is large enough (i.e., the shipment contains at least 100 switches) to satisfy the rule of thumb.

  3. We have seen at least one probability histogram for a binomial distribution - Free-Throw Freddy and his 5 shots. The resulting histogram in that case was for B(5, 0.7), but recall that to get the appropriate probabilities from Table C, we had to look at B(5, 0.3). Select other values for n and p (ones that you can use Table C to get) and plot the resulting probability distribution B(n,p) until you feel confident you know what you're doing. Use this applet to compare your histograms to the correct answers.

  4. Return to the binomial applet to experiment with it some more. Pick a value of p and stick with it while you vary the value of n. What do you notice about the center and spread of the distribution as you vary n? Does this seem to agree with what the formulas for mean and standard deviation tell you (see bottom of p. 380)?

    The formulas indicate that both mean and s.d. will grow if n grows (p staying fixed). You should observe this, although it would be easy to think you had not until you observe that a change in n also may result in a change in horizontal scale.

  5. In a binomial setting (see p. 376) where we assume samples of size n, the two most common random variables are the count of “successes” and the proportion of successes. What is the distinction between these two random variables? How are they related?

    The count and proportion are just the frequency and relative frequency of successes respectively. The proportion is obtained from the count by dividing by n, a linear transformation. Since the effect of a linear transformation is simply to translate and contract/expand a distribution, one should expect the sampling distribution for sample proportions to have the same general shape as the sampling distribution for counts. Moreover, the scenarios in which it is valid to approximate the binomial distribution B(n,p) by a normal distribution ought to be the same in which it is valid to approximate the sampling distribution for proportions by a normal distribution.