Download Calculating Confidence Intervals for Population Means and more Study notes Statistics in PDF only on Docsity! 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(X̄) = µ, SD(X̄) = 100/√100 = 10 • X̄ → µ (Law of Large Numbers) • X̄ ∼̇ N(µ, 10) (CLT) 2 Recall from the previous lecture that X̄ ∼̇ 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 < X̄ < µ + 1.96 σ√ n ) = 0.95 Rearranging terms, we have P ( X̄ − 1.96 σ√ n < µ < X̄ + 1.96 σ√ n ) = 0.95 In other words, there is 95% probability that the random interval( X̄ − 1.96 σ√ n , X̄ + 1.96 σ√ n ) will cover µ. In our example, x̄ = 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) 3 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 1. Take a random sample of size n and calculate the sample mean x̄. 2. If n is large enough, x̄ ∼̇ N ( µ, σ√ n ) (by the CLT). 3. The confidence interval is given by ( x̄− 1.96 σ√ n , x̄ + 1.96 σ√ n ) 4 Interpretation of Confidence Intervals Suppose we repeat the following procedure multiple times: 1. Draw a random sample of size n 2. 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 x̄ = 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 (x̄− 1.96 σ√ n , x̄ + 1.96 σ√ n ). • 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 x̄± z∗ σ√ n 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. 7 Finding z∗ For a given confidence level (1− α), how do we find z∗? Let Z ∼ N(0, 1): 0.025 0.025 0.95 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 2.576 8
