Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Foundations of statistics wikipedia , lookup
Bootstrapping (statistics) wikipedia , lookup
Psychometrics wikipedia , lookup
History of statistics wikipedia , lookup
Taylor's law wikipedia , lookup
Confidence interval wikipedia , lookup
Resampling (statistics) wikipedia , lookup
|PART FIVE Essentials|--Interval Estimation--Large Samples Four Types of Problems Means--one-group; two-group Columns one and two of the four-column formula sheet Proportions--one-group; two-group Columns three and four of the four-column formula sheet Confidence level (confidence coefficient)--the probability that a confidence interval will actually contain the population parameter being estimated (confidence interval is a range of values that is likely to contain the population parameter being estimated). 90%, 95%, and 99% are the most popular, and correspond to z-values of 1.645, 1.960, and 2.576, respectively. Of these, 95% is the most popular, and is assumed unless information to the contrary is provided. Error (uncertainty) factors express precision, as in 40 ± 3. Upper confidence limit--the point estimate plus the error factor, 43 in this example Lower confidence limit--the point estimate minus the error factor, 37 in this example Error factor is the product of the relevant z-value and the standard error: zt * σxbar Required sample sizes for desired precision may be computed Increased precision means a lower error factor. Precision can be increased by increasing the sample size, n. Increasing n lowers the standard error, since the standard error = σ / n. Taken to the extreme, every member of the population may be sampled, in which case the error factor becomes zero--no uncertainty at all--and the population parameter is determined exactly. Economic considerations--the high cost of precision The required increase in n is equal to the square of the desired increase in precision. To double the precision--to cut the error factor in half--the sample size must be quadrupled. Doubling the precision may thus quadruple the cost. To triple the precision--to cut the error factor to 1/3 of its previous value, n must be multiplied by 9. Hypothesis Testing--Large Samples Four Types of Problems--Four-column formula sheet Means--one-group; two-group Proportions--one-group; two-group Null (Ho) and alternate (Ha) hypotheses Means, one-group Ho: μ = some value Ha: μ ≠ that same value (two-sided test) μ > that same value (one-sided test, high end, right side) μ < that same value (one-sided test, low end, left side) Means, two-group Ho: μ1 = μ2 Ha: μ1 ≠ μ2 (two-sided test) μ1 > μ2 (one-sided test, high end, right side) μ1 < μ2 (one-sided test, low end, left side) Proportions, one-group Ho: π = some value Ha: π ≠ that same value (two-sided test) π > that same value (one-sided test, high end, right side) π < that same value (one-sided test, low end, left side) Proportions, two-group Ho: π1 = π2 Ha: π1 ≠ π2 (two-sided test) π1 > π2 (one-sided test, high end, right side) π1 < π2 (one-sided test, low end, left side) Type I error Erroneous rejection of a true Ho Probability of a Type I error is symbolized by α. Type II error Erroneous acceptance of a false Ho Probability of a Type II error is symbolized by β. Selecting α--based on researcher’s attitude toward risk α--the researcher's maximum tolerable risk of committing a type I error 0.10, 0.05, and 0.01 are the most commonly used. Of these, 0.05 is the most common--known as "the normal scientific standard of proof." Table-z (critical value); symbolized by zt; determined by the selected α value α 2-sided z 1-sided z 0.10 1.645 1.282 0.05 1.960 1.645 0.01 2.576 2.326 Calculated-z (test statistic); symbolized by zc Fraction—(5th row of formula sheet) "signal-to-noise" ratio Numerator ("signal")--strength of the evidence against Ho Denominator ("noise")--uncertainty factor for the numerator Rejection criteria Two-sided test: |zc| >= |zt|; also p <= α One-sided test: |zc| >= |zt|, AND zc and zt have the same sign; also p <= α Significance level (p-value) ("p" stands for probability) Actual risk (probability) of a Type I error if Ho is rejected on the basis of the experimental evidence Graphically, the area beyond the calculated z-value, zc. Terminology--explain each of the following: inferential statistics, sample mean, population mean, estimator, estimate, unbiased estimator, point estimate, interval estimate, confidence interval, degree of confidence, confidence level, table-z, error factor, required sample size, upper confidence limit, lower confidence limit, hypothesis test, null hypothesis, alternate hypothesis, type I error, α, type II error, β, calculated-z (test statistic), critical region, table-z (critical value of z), rejection of the null hypothesis, non-rejection of the null hypothesis, p-value, hypothesis-test conclusion, independent samples, standard error of the difference, sample proportion, population proportion, pooled proportion (two-group proportion cases) Skills and Procedures given appropriate data, conduct estimation and hypothesis testing on the population mean of one group, involving these steps: make a point estimate of a population mean compute the sampling standard deviation (standard error) of the sample means compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence levels determine the sample size needed to obtain a given desired error factor state the null and alternate hypotheses regarding the population mean determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01 compute the calculated-z (test statistic) draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z (critical value) and the calculated-z (test statistic) interpret the conclusion determine and interpret the p-value given appropriate data, conduct estimation and hypothesis testing on the population means of two groups, involving these steps: make a point estimate of the difference between population means compute the sampling standard deviation (standard error) of the difference between sample means compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence levels determine the sample size needed to obtain a given desired error factor state the null and alternate hypotheses regarding the difference between population means determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01 compute the calculated-z (test statistic) draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z and the calculated-z interpret the conclusion determine and interpret the p-value given appropriate data, conduct estimation and hypothesis testing on the population proportion of one group, involving these steps: make a point estimate of a population proportion compute the sampling standard deviation (standard error) of the sample proportions compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence levels determine the sample size needed to obtain a given desired error factor state the null and alternate hypotheses regarding the population proportion determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01 compute the calculated-z (test statistic) draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z and the calculated-z interpret the conclusion determine and interpret the p-value given appropriate data, conduct estimation and hypothesis testing on the population proportions of two groups, involving these steps: make a point estimate of the difference between population proportions compute the sampling standard deviation (standard error) of the difference between sample proportions compute and interpret the error factor for the interval estimate for the 90%, 95% and 99% confidence levels determine the sample size needed to obtain a given desired error factor state the null and alternate hypotheses regarding the difference between population proportions determine the table-z (critical value of z) for alpha levels of 0.10, 0.05 and 0.01 compute the calculated-z (test statistic) draw the appropriate hypothesis-test conclusion based on the given level of α, the table-z and the calculated-z interpret the conclusion determine and interpret the p-value Concepts- explain why a confidence interval becomes larger as the confidence level increases explain why a confidence interval becomes smaller as the sample size increases describe the nature of the trade-off between precision and cost identify the type of error that is made if the null hypothesis is "the defendant is innocent," and an innocent defendant is erroneously convicted identify the type of error that is made if the null hypothesis is "the defendant is innocent," and a guilty defendant is erroneously acquitted explain why a researcher seeking to reject a null hypothesis may tend to prefer a one-sided alternative hypothesis