14 Critical Thinking Since is a random variable, so are the endpoints After the confidence interval is numerically fixed for a specific sample, it either does or does not contain µ.
15 Critical ThinkingIf we repeated the confidence interval process by taking multiple random samples of equal size, some intervals would capture µ and some would not!Equation states that the proportion of all intervals containing µ will be c.
17 Estimating µ When σ is Unknown In most cases, researchers will have to estimate σ with s (the standard deviation of the sample).The sampling distribution for will follow a new distribution, the Student’s t distribution.
20 The t DistributionUse Table 6 of Appendix II to find the critical values tc for a confidence level c.The figure to the right is a comparison of two t distributions and the standard normal distribution.
21 Using Table 6 to Find Critical Values Degrees of freedom, df, are the row headings.Confidence levels, c, are the column headings.
22 Maximal Margin of Error If we are using the t distribution:
25 Estimating p in the Binomial Distribution We will use large-sample methods in which the sample size, n, is fixed.We assume the normal curve is a good approximation to the binomial distribution if both np > 5 and nq = n(1-p) > 5.
32 Choosing Sample SizesWhen designing statistical studies, it is good practice to decide in advance:The confidence levelThe maximal margin of errorThen, we can calculate the required minimum sample size to meet these goals.
33 Sample Size for Estimating μ If σ is unknown, use σ from a previous study or conduct a pilot study to obtain s.Always round n up to the next integer!!
34 Sample Size for Estimating If we have no preliminary estimate for p, use the following modification:
35 Independent SamplesTwo samples are independent if sample data drawn from one population is completely unrelated to the selection of a sample from the other population.Occurs when we draw two random samples
36 Dependent SamplesTwo samples are dependent if each data value in one sample can be paired with a corresponding value in the other sample.Occur naturally when taking the same measurement twice on one observationExample: your weight before and after the holiday season.
37 Confidence Intervals for μ1 – μ2 when σ1, σ2 known
38 Confidence Intervals for μ1 – μ2 when σ1, σ2 known
44 Summarizing Intervals for Differences in Population Means
45 Estimating the Difference in Proportions We consider two independent binomial distributions.For distribution 1 and distribution 2, respectively, we have:n1 p1 q1 r1n2 p2 q2 r2We assume that all the following are greater than 5: