Chapter 8 Estimation Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania

Estimating µ When σ is Known

Point Estimate An estimate of a population parameter given by a single number.

Margin of Error Even if we take a very large sample size, will differ from µ.

Confidence Levels A confidence level, c, is any value between 0 and 1 that corresponds to the area under the standard normal curve between –zc and +zc.

Critical Values

Common Confidence Levels

Recall From Sampling Distributions If we take samples of size n from our population, then the distribution of the sample mean has the following characteristics:

A Probability Statement In words, c is the probability that the sample mean will differ from the population mean by at most

Maximal Margin of Error Since µ is unknown, the margin of error | - µ| is unknown. Using confidence level c, we can say that differs from µ by at most:

Confidence Intervals

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.

Critical Thinking If 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.

Multiple Confidence Intervals

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.

The t Distribution

The t Distribution

The t Distribution Use 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.

Using Table 6 to Find Critical Values Degrees of freedom, df, are the row headings. Confidence levels, c, are the column headings.

Maximal Margin of Error If we are using the t distribution:

What Distribution Should We Use?

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.

Point Estimates in the Binomial Case

Margin of Error The magnitude of the difference between the actual value of p and its estimate is the margin of error.

The Distribution of The distribution is well approximated by a normal distribution.

A Probability Statement With confidence level c, as before.

Public Opinion Polls

Choosing Sample Sizes When designing statistical studies, it is good practice to decide in advance: The confidence level The maximal margin of error Then, we can calculate the required minimum sample size to meet these goals.

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!!

Sample Size for Estimating If we have no preliminary estimate for p, use the following modification:

Independent Samples Two 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

Dependent Samples Two 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 observation Example: your weight before and after the holiday season.

Confidence Intervals for μ1 – μ2 when σ1, σ2 known

Confidence Intervals for μ1 – μ2 when σ1, σ2 known

Confidence Intervals for μ1 – μ2 when σ1, σ2 unknown If σ1, σ2 are unknown, we use the t distribution (just like the one-sample problem).

What if σ1 = σ2 ? If the sample standard deviations s1 and s2 are sufficiently close, then it may be safe to assume that σ1 = σ2. Use a pooled standard deviation. See Section 8.4, problem 27.

Summarizing Intervals for Differences in Population Means

Estimating the Difference in Proportions We consider two independent binomial distributions. For distribution 1 and distribution 2, respectively, we have: n1 p1 q1 r1 n2 p2 q2 r2 We assume that all the following are greater than 5:

Estimating the Difference in Proportions

Critical Thinking