1. Chapter 10: Basics of Confidence Intervals
December 18
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10: Intro to Confidence Intervals

2. In Chapter 10: 10.1 Introduction to Estimation
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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
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Basic Biostat

3. 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).
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4. 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).
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5. 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).
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6. 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)
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7. (1 – α)100% Level of Confidence
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8. 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
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9. (1 – α)100% CI for µ
margin of error m
where:
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10. 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
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11. 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
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12. How 95% CIs behave
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13. 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
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14. 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
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15. 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
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16. 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
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17. 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
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