|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
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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-
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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