chapter 10 statistics

Confidence Interval Estimator of μ:

The probability 1-α is called the confidence level. E.g. 1-α=0.95, then called the 95%confidence interval estimator of μ.

Information and the Width of the Interval:

The width of the confidence interval estimate is a function of the population standard deviation, the confidence level, and the sample size. -Doubling the population standard deviation has the effect of doubling the width of the confidence interval estimate. -Decreasing the confidence level narrows the interval; increasing it widens the interval. However, a large confidence level is generally desirable because that means a larger proportion of confidence interval estimates that will be correct in the long run. 95% confidence is considered "standard." -With a larger sample, the interval will become smaller.

Interpret the Confidence Interval Estimate:

Wrong way to interpret the confidence interval estimate in Example 10.1: there is a 95% probability that the population mean lies between 340.76 and 399.56. In fact, the population mean is a fixed but unknown quantity.

Consistency:

an unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.

Unbiased Estimator:

an unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter.

Point Estimator and Drawbacks

draws inferences about a population by estimating the value of an unknown parameter using a single value or point. Drawbacks: 1. It is virtually certain that the estimate will be wrong (The probability that a continuous random variable will equal a specific value is 0; that is, the probability that 𝑋 will exactly equal μ is 0.) 2. We often need to know how close the estimator is to the parameter. 3. The point estimators do not have the capacity to reflect the effects of larger sample sizes.

Interval Estimator:

draws inferences about a population by estimating the value of an unknown parameter using an interval. (affected by the sample size)

Relative Efficiency:

if there are two unbiased estimators of a parameter, the one whose variance is smaller is said to have relative efficiency. The sample mean is relatively more efficient than the sample median when estimating the population mean.

Sampling error

is the difference between the sample and the population that exists only because of the observations that happened to be selected for the sample.