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Introduction to Inference
Confidence Intervals
- A Typical Inference Problem
- The 95% Confidence Interval
- Definition
- Calculating a 95% Confidence Interval
- Interpretation of Confidence Intervals
- The General Form of a Confidence Interval
- Finding z∗
- Factors Affecting CI Length
1
A Typical Inference Problem
Suppose we want to find out about the mean lifetime μ of a certain brand of light bulbs. Suppose that the true mean μ is unknown, but we know (perhaps from previous studies) that the SD σ of the light bulb lifetime is 100 hours. In order to estimate the population mean μ we:
- Take a SRS of 100 light bulbs.
- Calculate the mean lifetime in the sample to be 1100 hours. What can we say about the population mean?
- E( X¯) = μ, SD( X¯) = 100/
- X¯ → μ (Law of Large Numbers)
- X¯ ∼˙ N (μ, 10) (CLT)
2 Recall from the previous lecture that
X¯ ∼˙ N (μ, √σ n
The distribution is exact if the population distribution is normal, and approximately correct for large n in other cases, by the CLT. Thus,
P
μ − 1. 96 √σ n
< X < μ¯ + 1. 96 √σ n
Rearranging terms, we have
P
X^ ¯ − 1. 96 √σ n < μ <^ X¯ + 1. 96 √σ n
In other words, there is 95% probability that the random interval ( X^ ¯ − 1. 96 √σ n ,^ X¯ + 1. 96 √σ n
will cover μ.
In our example, ¯x = 1100, σ = 100, and n = 100. Therefore, the 95% confidence interval for μ is
(1100 − 1. 96 × 10 , 1100 + 1. 96 × 10) = (1080. 4 , 1119 .6)
Calculating a 95% Confidence Interval
For the time being, we’ll continue to assume that σ is known. To calculate a 95% confidence interval for the population mean μ
- Take a random sample of size n and calculate the sample mean ¯x.
- If n is large enough, ¯x ∼˙ N
μ, √σn
(by the CLT).
- The confidence interval is given by ( x ¯ − 1. 96 √σ n ,^ x¯^ + 1.^96
√σ n
Interpretation of Confidence Intervals
Suppose we repeat the following procedure multiple times:
- Draw a random sample of size n
- Calculate a 95% confidence interval for the sample
95% of the intervals thus constructed will cover the true (unknown) population mean.
5
Example Consider estimating the speed of light using 64 measurements with sample mean ¯x = 298, 054 km/s. Assume we know (from previous experience) that the SD of measurements made using the same procedure is 60 km/s. What is a 95% CI for the true speed of light? Incorrect:
- There is a 95% probability that the true speed of light lies in the interval (298,039.3, 298,068.7).
- In 95% of all possible samples, the true speed of light lies in the interval (298,039.3, 298,068.7). Correct:
- There is 95% confidence that the true speed of light lies in the interval (298,039.3, 298,068.7).
- There is 95% probability that the true speed of light lies in the random interval (¯x − 1. 96 √σn , ¯x + 1. 96 √σn ).
- If we repeatedly draw samples and calculate confidence intervals using this procedure, 95% of these intervals will cover the true speed of light.
6 General Form of a Confidence Interval
In general, a CI for a parameter has the form
estimate ± margin of error
where the margin of error is determined by the confidence level (1 − α), the population SD σ, and the sample size n.
A (1 − α) confidence interval for a parameter θ is an interval computed from a SRS by a method with probability (1 − α) of containing the true θ.
For a random sample of size n drawn from a population of unknown mean μ and known SD σ, a (1 − α) CI for μ is
x¯ ± z∗^ √σn
Here z∗^ is the critical value, selected so that a standard Normal density has area (1 − α) between −z∗^ and z∗. The quantity z∗σ/√n, then, is the margin error.
If the population distribution is normal, the interval is exact. Otherwise, it is approximately correct for large n.
Finding z∗ For a given confidence level (1 − α), how do we find z∗? Let Z ∼ N (0, 1):
0.025 0.
0.
P (−z∗^ ≤ Z ≤ z∗) = (1 − α) ⇐⇒ P (Z < −z∗) = α 2
Thus, for a given confidence level (1 − α), we can look up the corresponding z∗^ value on the Normal table. Common z∗^ values: Confidence Level 90 95 99 z∗^ 1.645 1.96 2.
