Unit 3: Inference for Categorical Data: Proportions

18 terms. Say what each one means, then open the card to check.

• Confidence interval

  A range of plausible values for a parameter, built as a point estimate plus or minus a margin of error from one random sample.

  Example. If 132 of a random sample of 300 cyclists say they always wear a helmet, a 95 percent interval for the share of all cyclists who do is about 0.384 to 0.496.

• Confidence level

  The long-run success rate of the method: the percent of intervals, over many random samples, that would capture the true parameter.

  Example. At 95 percent confidence, about 95 of every 100 intervals built the same way from different random samples would capture the true proportion.

• Margin of error

  The critical value times the standard error. It is half the width of the interval and describes how far the estimate is likely to be from the parameter at the stated confidence level.

  Example. An interval of 0.41 to 0.49 has a center of 0.45 and a margin of error of 0.04.

• Standard error

  An estimate of a statistic's standard deviation, found by putting sample values into the formula, such as the square root of p-hat(1 - p-hat)/n.

  Example. With 132 successes out of 300, the standard error of the sample proportion is the square root of 0.44(0.56)/300, about 0.029.

• Null hypothesis

  The claim that a parameter equals a specific value, often a value of no effect or no difference. A test assumes it is true and weighs the evidence against it.

  Example. H0: p = 0.25, where p is the proportion of all students at a school who walk to class.

• Alternative hypothesis

  The claim the study looks for evidence for. It uses less than, greater than or not equal to, and it is chosen before seeing the data.

  Example. A principal who suspects more than a quarter of the students walk to class writes Ha: p > 0.25.

• Test statistic

  A standardized measure of how far the sample result is from the null value, such as z = (p-hat - p0) divided by the standard deviation under the null.

  Example. A sample proportion of 0.38 against a null value of 0.32 with n = 300 gives z of about 2.23.

• P-value

  The probability, assuming the null hypothesis is true, of getting a sample result at least as extreme as the one observed, in the direction of the alternative.

  Example. A P-value of 0.03 means that if the null were true, only about 3 percent of random samples would give a result this extreme or more.

• Significance level

  The cutoff alpha, chosen before the study: reject the null hypothesis when the P-value is less than alpha. It is also the probability of a Type I error.

  Example. With alpha = 0.05, a P-value of 0.021 leads to rejecting the null, but a P-value of 0.31 does not.

• Type I error

  Rejecting the null hypothesis when it is actually true: finding an effect that is not there.

  Example. A school bans a snack after concluding that more than 10 percent of students are allergic to it, when the rate really is 10 percent.

• Type II error

  Failing to reject the null hypothesis when the alternative is actually true: missing an effect that is there.

  Example. A school fails to find that more than 30 percent of its buses run late, when in fact more than 30 percent do.

• Power

  The probability that a test rejects the null hypothesis when a particular alternative value is true. Power equals 1 minus the probability of a Type II error.

  Example. If a test has power 0.70 against p = 0.45, it would miss a true proportion of 0.45 about 30 percent of the time.

• Pooled proportion

  The combined proportion of successes in both samples, x1 + x2/n1 + n2, used in the standard error when testing that two proportions are equal.

  Example. With 90 of 150 successes in one group and 70 of 150 in the other, the pooled proportion is 160/300, about 0.533.

• Difference in proportions

  The parameter p1 - p2 that compares two populations or two treatments. An interval that contains 0 does not show a difference.

  Example. An interval of -0.02 to 0.12 for p-north minus p-south contains 0, so it does not show that the two regions differ.

• Expected count

  The count a cell would have if the null hypothesis were exactly true: its row total times its column total, divided by the grand total.

  Example. A cell in a row of 150 and a column of 80 in a table of 400 has expected count 150(80)/400 = 30.

• Chi-square statistic

  The sum over all cells of (observed - expected) squared, divided by expected. Larger values mean the counts are further from what the null predicts.

  Example. A two-row, four-column table of club choices gives a chi-square statistic of about 7.9 with 3 degrees of freedom.

• Chi-square test for homogeneity

  A test of whether the distribution of one categorical variable is the same across several populations or treatments, using separate samples or groups.

  Example. Separate random samples of 80 patients at each of four clinics, compared on whether each patient kept a follow-up visit.

• Chi-square test for independence

  A test of whether two categorical variables are associated in one population, using a single sample classified by both variables.

  Example. One random sample of 250 adults, each classified by whether they own a pet and by the kind of home they live in.