compare the Z-distribution with Student’s t-distribution (t-distribution), focusing on the four attributes of probability distributions

1. Considering the definition of “point estimate,” used in determination of an “interval estimate” (construction of a confidence interval) for an unknown population parameter, determine just the numerical value of the point estimate for a sample of your choice when

1. an “interval estimate” for the unknown population mean is of interest.
2. an “interval estimate” for the unknown population proportion is of interest.

NOTE: You may consider either one sample or two different samples with the size(s) of your choice.  For the proportion case you may consider gender, if people are involved in your sample; or, you may consider “defective” and “non-defective” if members of your sample are, say, calculators.

2. When you construct a 95% confidence interval, what are you 95% confident about?

3. All other things remaining the same, is a 90% confidence interval narrower or wider than a 95% confidence interval? Briefly and clearly explain your answer.  HINT: Think of the corresponding margins of error (or error bounds).

4. All other things remaining the same, what is the impact of an increase in sample size on the width of the confidence interval? Explain your answer briefly and clearly

5. Briefly,

1. compare the Z-distribution with Student’s t-distribution (t-distribution), focusing on the four attributes of probability distributions (random variable name, behavior [curve], mean, and standard deviation).
2. compare the format of the Z-table with that of the t-table, focusing on the fundamental differences in format and the information involved.

6. Briefly answer the following questions:

1.  When should we use the t-table, instead of the Z-table, while constructing confidence intervals or conducting tests of hypotheses.
2. When must the requirement of the normality (or approximate normality) of the population distribution be met when we use Z or t?
3. What would be the impact on the confidence interval, if a mistake is made and Z table is used instead of the t-table?  Support your answer with a brief description and/or computation.  HINT: Examine the superimposed graphs of Z-distribution and t-distribution given in Lane, and focus on the relative magnitudes of Za/2 and ta/2, each being a factor in the margin of error (error bound).

7. (Show your work.) In a recent Zogby International Poll, nine of 48 respondents rated the likelihood of a terrorist attack in their community as “likely” or “very likely.”

1.  Use the “plus four” method, as explained by Illowsky, to create a 97% confidence interval for the proportion of American adults who believe that a terrorist attack in their community is likely or very likely.
2.  Explain what this confidence interval means in the context of the problem.

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