Confidence Intervals in Statistics: Calculation and Interpretation, Study notes of Statistics

Download Confidence Intervals in Statistics: Calculation and Interpretation and more Study notes Statistics in PDF only on Docsity! Confidence Interval Review Reading: Sections 7.1-7.3 Spring 2009 Statistics 428 1 Confidence Interval Ideas • CIs are ranges of values that are reasonable estimates (guesses) for population parameters – CIs are ranges of parameter values that are consistent with the observed data • CIs are constructed using probability distributions for unobserved sample statistics (random variables) – The probability that an unobserved interval (where the limits are Spring 2009 Statistics 428 2 random variables) captures the true population parameter value is the confidence of the interval = 1-α. – The probability that an unobserved interval fails to capture the true population parameter value is the significance of the interval = α. • CIs can be – two-sided (half of the significance at high and low ends of the distribution) – one-sided (all of significance at one end of the distribution) Wider and Narrower For most confidence intervals: • To make a confidence interval wider: – Increase confidence (1-α) = decrease significance (α) (be willing to make mistakes less often) – Decrease the sample size (n) Spring 2009 Statistics 428 5 • To make a confidence interval narrower: – Decrease confidence (1-α) = increase significance (α) (be willing to make mistakes more often) – Increase the sample size (n) Sample Size • Rearrange the formula for a confidence interval to find the sample size necessary to limit the margin of error (half the width of a symmetric confidence interval) to a Spring 2009 Statistics 428 6 certain size at a given significance α. – For the CI for the mean (µ) of a Normal distribution with known population standard deviation (σ): Confidence Intervals for Common Distributions Spring 2009 Statistics 428 7 Solution: Using the strategy: 1. Determine the distribution of the data 2. Choose a statistic We only have one observation, so there’s really only one choice of statistic: Y. 3. Using the data distribution, find the distribution of the chosen statistic 4. Using the CDF for the statistic, find limits c and d Spring 2009 Statistics 428 10 Put half the error below c and half above d Use the properties of what we know must be a square triangle at the bottom and top of the distribution. Area = ½height*width = ½width2 for right triangles (as we have here) The lower part: The upper part: Put it together: Spring 2009 Statistics 428 11 5. Solve for τ in the center of the interval, and statistics a and b in the limits Confidence Interval: Suppose you observe Y=10 watts. Calculate (and interpret) a 96% confidence interval for the average total watts (τ). I am 96% confident that the true average power used in total by both of the two electrical devices is between 9.2 and 10.8 watts. Spring 2009 Statistics 428 12 Would the width of this interval likely increase or decrease if we observed more than 100 instances of power use? DECREASE Would the width of this interval increase or decrease if we increased the confidence to 98%? INCREASE