Math 143 C/E, Spring 2001
IPS Reading Questions
Chapter 8, Section 1 (pp. 584-588; up to “Significance test for a single proportion”)



  1. For both sample means and sample proportions the true margin of error for a given level C of confidence is z*sstatistic, where, in the case of , sstatistic = s/n1/2, and in the case of , sstatistic = [p(1-p)/n]1/2. Nevertheless, for proportions, you are told to use the standard error SE = [(1 - )/n]1/2 instead of s, though this only leads to an approximate margin of error (and hence, only to an approximate level C confidence interval). Explain why this approach, though less accurate, leads to a more realistically-computable confidence interval than did our method for sample means presented in Section 6.1.

    For the method of 6.1, computation of the margin of error requires knowledge of the value of s (in order to compute s, the population's standard deviation. If we do not know the population mean, why would we know it's spread? In fact, generally we will not know s in practice, making this procedure a little unrealistic. Similarly, the value of s, would require knowledge of p, and if we had p we probably would not have needed to get the sample in the first place. Using SE in its place allows us to get nearly the correct margin of error, and we only need to know the sample proportion to compute it.

  2. Under what circumstances is the approximate confidence interval for proportions, described on p. 587, going to be close enough to the true confidence interval to be satisfactory?

    It will be close enough when the ratio of the population size to the sample size is at least 10, and when there are at least 10 successes and 10 failures in the sample. (Isn't it convenient that most of the sizes required by the rules of thumb we have seen are 10?)

  3. We employ confidence intervals because we know that the sample we gather may yield a sample statistic whose value is somewhat different than the population parameter we seek. What sources of error in sampling does this procedure account for?

    The main assumption underlying the computation of the margin of error is that the sample is a true SRS. If so, this means that the only variation we would see from one sample to another is the product of random chance in the sample selection process. The value of the margin of error reflects the low probability of obtaining certain types of samples (ones whose sample statistic is far from the population parameter) and the high probability of obtaining others (whose sample statistic is close to the parameter). Given this, if our sample is not an SRS, if the selection process we use somehow (perhaps through bias, but perhaps not) throws off these probabilities (making, say, a more extreme sample more likely of being selected) then the confidence interval will not have taken these altered probabilities into account and will not be accurate.