14Critical 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.
15Critical 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.
17Estimating 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.
20The 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.
21Using Table 6 to Find Critical Values Degrees of freedom, df, are the row headings.Confidence levels, c, are the column headings.
22Maximal Margin of Error If we are using the t distribution:
25Estimating 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.
32Choosing 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.
33Sample 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!!
34Sample Size for Estimating If we have no preliminary estimate for p, use the following modification:
35Independent 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
36Dependent 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.
37Confidence Intervals for μ1 – μ2 when σ1, σ2 known
38Confidence Intervals for μ1 – μ2 when σ1, σ2 known
44Summarizing Intervals for Differences in Population Means
45Estimating 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: