Chapter 10: Basics of Confidence Intervals December 18 12/31/2018Basic Biostat 10: Intro to Confidence Intervals

In Chapter 10: 10.1 Introduction to Estimation 12/31/2018 In Chapter 10: 10.1 Introduction to Estimation 10.2 Confidence Interval for μ (σ known) 10.3 Sample Size Requirements 10.4 Relationship Between Hypothesis Testing and Confidence Intervals 12/31/2018 Basic Biostat

Statistical Inference Recall the goal of statistical inference: using statistics calculated in the sample We want to learn about population parameters… Statistical inference is the logical process by which we make sense of numbers. It is how we generalize from the particular to the general. This is an important scientific activity. Consider a study that wants to learn about the prevalence of a asthma in a population. We draw a SRS from the population and find that 3 of the 100 individuals in the sample have asthma. The prevalence in the sample is 3%. However, are still uncertain about the prevalence of asthma in the population; the next SRS from the same population may have fewer or more cases. How do we deal with this element of chance introduced by sampling? A similar problem occurs when we do an experiment Each time we do an experiment, we expect different results due to chance. How do we deal with this chance variability? Our first task is to distinguish between parameters and statistics. Parameters are the population value. Statistics are from the sample. The statistics will vary from sample to sample: they are random variables. In contrast, the parameters are invariable; they are constants. To help draw this distinction, we use different symbols for parameters and statistics. For example, we use “mu” to represent the population mean (parameter) and “xbar” to represent the sample mean (statistic). 12/31/2018

Recall: We introduce estimation concepts with the The distinction between a sample statistic (e.g., sample mean “x-bar”) and population parameter (e.g., population mean µ) The two forms of statistical inference: Hypothesis testing (Introduced in Ch 9) Estimation (Introduced in this chapter) We introduce estimation concepts with the the one-sample z CI for µ Statistical inference is the logical process by which we make sense of numbers. It is how we generalize from the particular to the general. This is an important scientific activity. Consider a study that wants to learn about the prevalence of a asthma in a population. We draw a SRS from the population and find that 3 of the 100 individuals in the sample have asthma. The prevalence in the sample is 3%. However, are still uncertain about the prevalence of asthma in the population; the next SRS from the same population may have fewer or more cases. How do we deal with this element of chance introduced by sampling? A similar problem occurs when we do an experiment Each time we do an experiment, we expect different results due to chance. How do we deal with this chance variability? Our first task is to distinguish between parameters and statistics. Parameters are the population value. Statistics are from the sample. The statistics will vary from sample to sample: they are random variables. In contrast, the parameters are invariable; they are constants. To help draw this distinction, we use different symbols for parameters and statistics. For example, we use “mu” to represent the population mean (parameter) and “xbar” to represent the sample mean (statistic). 12/31/2018

Estimating µ, σ known Objective: to estimate the value of population mean µ under these conditions Simple Random Sample (SRS) Population Normal or large sample The value of σ is known The value of μ is NOT known Statistical inference is the logical process by which we make sense of numbers. It is how we generalize from the particular to the general. This is an important scientific activity. Consider a study that wants to learn about the prevalence of a asthma in a population. We draw a SRS from the population and find that 3 of the 100 individuals in the sample have asthma. The prevalence in the sample is 3%. However, are still uncertain about the prevalence of asthma in the population; the next SRS from the same population may have fewer or more cases. How do we deal with this element of chance introduced by sampling? A similar problem occurs when we do an experiment Each time we do an experiment, we expect different results due to chance. How do we deal with this chance variability? Our first task is to distinguish between parameters and statistics. Parameters are the population value. Statistics are from the sample. The statistics will vary from sample to sample: they are random variables. In contrast, the parameters are invariable; they are constants. To help draw this distinction, we use different symbols for parameters and statistics. For example, we use “mu” to represent the population mean (parameter) and “xbar” to represent the sample mean (statistic). 12/31/2018

Estimating µ Two forms of estimation Point estimation ≡ most likely value of parameter µ (i.e., sample mean x-bar) Interval estimate ≡ the sample mean is surrounded with a margin of error to create a confidence interval (CI) 12/31/2018

(1 – α)100% Level of Confidence 12/31/2018

Common Levels of Confidence Confidence level 1 – α Alpha level α Z value z1–(α/2) .90 .10 1.645 .95 .05 1.960 .99 .01 2.576 12/31/2018

(1 – α)100% CI for µ margin of error m where: 12/31/2018

95% CI for µ (Example) Body weights of 20-29-year-old males Unknown μ, σ = 40 SRS of n = 64  calculate x-bar =183 margin of error m We have 95% confidence µ is in this interval 12/31/2018

99% CI for µ (Example) Body weights of 20-29-year-old males Unknown μ, σ = 40 SRS of n = 64  sample mean x-bar =183 margin of error m We have 99% confidence µ is in this interval 12/31/2018

How 95% CIs behave 12/31/2018

How CIs behave (cont.) The curve represents the sampling distribution of the mean Five 95% CIs for µ from the sampling distribution are shown below the curve The third CI failed to capture μ 12/31/2018

Sample Size Requirements How large a sample is need for a (1 – α)100% CI for µ with margin of error m? Examples (next slide) use these assumptions: σ = 40 95% confidence  z1–.05/2 = z.975 = 1.96 Varying m 12/31/2018

Examples: Sample Size Requirements Round-up to next integer so m no greater than stated Smaller m requires larger n Square root law: quadruple n to double precision 12/31/2018

10.4 Relation Between Testing and Confidence Intervals Rule: Reject H0 at the α level for significance when μ0 falls outside the (1−α)100% CI. Illustrations: Next slide 12/31/2018

Example: Testing and CIs Illustration: Test H0: μ = 180 This CI excludes 180 Reject H0 at α =.05 Retain H0 at α =.01 This CI includes 180 12/31/2018