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Chapter 21
Estimating
● Statistical Inference
○ Drawing conclusions about a population from sample data
● One kind of conclusion answers questions such as the following
○ “What percent of employed women have a college degree?”
○ “What is the mean survival time for patients with a given type of cancer?”
■ Can’t feasibly study all individuals in the population , so take a sample Z
● Parameters
○ Numbers that describe a population
○ To estimate a population parameter, use a statistic, a number calculated from the sample, as your
estimate
● Does it make sense to use this statistic to represent the population?
○ Yes - it’s our best guess
● But, still only an estimate
● If you’ve done everything right, then
○ The sample statistic ≈ the true population proportion
● Since it’s an estimate, how can we reflect the degree of our uncertainty about this?
● C% Confidence Interval
○ A range of value calculated from sample data by a process that’s designed to capture the true
population parameter in C% of all samples
■ Assuming you repeat sampling ot infinity
○ C is often 95 (as in 95% confidence interval)
Estimating with Confidence
● Goal
○ Estimate the proportion p of the individuals in a population who have some characteristics (say,
“success”)
● p̂
○ Proportion of that characteristic (“success”) in a simple random sample (SRS)
● How accurate is the statistic p̂ as an estimate of the parameter p?
● Frequentists will think to themselves
○ “What if we sampled and generated p̂ an infinite number of times?”
● Variance of p̂
○ Clear pattern in the long run
○ Described by a normal curve
● Sampling distribution
○ The distribution of values taken by a statistic in
■ All possible samples
■ Of the same size (n)
■ From the same population
● P-hat
○ The sample proportion of successes
● If the sample size is large enough
○ The sampling distribution of p̂ is approximately normal
○ The mean of the sampling distribution is p
○ The standard deviation of the sampling distribution:
, ● If we’re sampling many, many times from the same population and samples are the same size
○ sampling distribution of the statistic has an approximate normal shape
○ The mean of sampling distribution of the statistic equals the true population parameter
○ The standard deviation of the sampling distribution is predictable
○ The variance of the statistic follows a regular pattern
■ Conceptually, this combines the knowledge of probability and frequentism with sampling
and inference
● Standard Error
○ The technical name we give for the standard deviation of the
sampling distribution of a sample statistic
○ What the statistic is a proportion, the formula is:
● Using the 68-95-99.7 rule, for any value of p
○ When the population proportion has the value p, 95% of all
samples catch p in the interval extending roughly 2 standard
errors on either side of p
○ Per the 68-95-99.7 rule, 95% of the distribution’s values fall
within 2 standard deviations on either side of the mean
○ The standard error is the standard deviation of the sampling distribution
■ Therefore ~95% of samples (the values in the sampling distribution) encompass the value
of the parameter (the mean of the sampling distribution), in the range of values given by
2 standard errors (the standard deviation of the sampling distribution)
● What’s wrong with what we just did?
○ Most of the time, we don’t actually know what p is
○ No way to verify exactly without actually sampling everyone
○ Solution: substitute p̂ from a sample to compute interval
● The confidence interval
○ From a SRS of size n from a large population contains an unknown proportion of p of successes
○ Call the proportion of successes in this sample p̂
● An approximate 95% confidence interval for the parameter p is p̂ ± 2 √ ❑
Understanding Confidence Intervals
● Confidence intervals are frequentist in nature
○ “What would happen if we repeated the sampling many, many times?”
● The 95% in a 95% confidence interval is a probability
○ The probability that the method produces an interval that captures the true parameter over 95 of
repeated samples
● The 95% refers to your confidence in the interval itself, not to any point estimate inside of itt
○ By this we mean, the probability that any actual number lies within a CI you’ve found is 1 or 0
(i.e., it does or it doesn’t)
■ Ex: CI [92, 95], probability that 4 falls in this CI is 100% or 1
● CIs aren’t exacts for two reasons
○ 1) the sampling distribution of the sample proportion p̂
■ Only approximately normal
○ 2) the standard error of p
■ Not exact because we used p̂ instead
○ This means we use
■ A new estimate of the standard deviation of the sampling distribution every time we take
a new sample, even though the true standard deviation never changes
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