Suppose X is a random variable with a t distribution with 4 degrees of freedom. What is P(X a) = 0.01?
Confidence interval for a mean: • Want to estimate μ (“population mean” or “true mean” or “mean from model in the theoretical world”). We don’t know μ. • Estimate it with the mean (or average) calculated from the data (the “sample mean”). We have a numerical value for this. • The confidence interval gives a range of plausible values for what μ could be, based on our observed data.
What is the effect on the confidence interval of quadrupling n?
Practice Problem: A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A 95% confidence interval for the mean wait time based on this sample is (128 minutes, 147 minutes), calculated using the t distribution. a) What numbers were needed to calculate this confidence interval? b) What does a 95% confidence level mean in this context? c) How could she decrease the width of the confidence interval without losing confidence?
Practice Problem: A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A 95% confidence interval for the mean wait time based on this sample is (128 minutes, 147 minutes), calculated using the t distribution. a) What numbers were needed to calculate this confidence interval?
Practice Problem: A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A 95% confidence interval for the mean wait time based on this sample is (128 minutes, 147 minutes), calculated using the t distribution. b) What does a 95% confidence level mean in this context?
Practice Problem: A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A 95% confidence interval for the mean wait time based on this sample is (128 minutes, 147 minutes), calculated using the t distribution. c) How could she decrease the width of the confidence interval without losing confidence?
Interpretation of a confidence interval: If we perform our data collection procedure a large number of times, and each time we use the data we collect to estimate something (for example, a proportion or mean), and each time we calculate a 95% confidence interval for what we’re trying to estimate, the confidence intervals for 95% of the samples of data will include the true value of what is being estimated as long as the necessary conditions for the confidence interval hold.
“Nominal” versus “Actual” Coverage of a Confidence Interval • The aim of a confidence interval is to capture the population mean or proportion with a given probability (e.g., 95%), that is, in 95% of the possible samples of data. • The confidence level (e.g., 95%) is often called the nominal coverage. • The actual coverage of a confidence interval is the actual probability that the interval contains the population mean or proportion. • The formulae for CIs rely on assumptions being true. If the assumptions are true, then nominal coverage = actual coverage
Necessary conditions for the confidence intervals we’ve learned: 1. The observations are independent. 2. The estimator has (approximately) a Normal distribution. • For estimating a proportion, the Central Limit Theorem ensures the estimator has approximately a Normal distribution for large n. In the video, we saw: o p=0.5, n=10 was not large enough (actual coverage