Statistics in Applied Science and Technology Chapter 8 Estimation of Population Means

Key Concepts in this Chapter Point estimate confidence intervals Pooled standard deviation Determination of sample size

Introduction to Estimation and Hypothesis Testing Estimation and hypothesis testing are primary tools of statistical inference. The unknown characteristic (parameter) of a population is usually estimated from a statistic computed from data of a sample. The purpose of statistical inference is to reach conclusions from our data (sample) and to support our conclusion (about population) with probability statements.

Point Estimates Two ways of estimating a population parameter: a point estimate and a confidence interval Point estimate - a single value that represents some unknown population characteristic, such as population mean. The weakness in the point estimate is that it fails to make a probability statement as to how close the estimate is to the population parameter.

Confidence Interval Confidence interval - a range of values that, with a known degree of certainty, includes an unknown population characteristic (parameter), such as population mean . 95% CI of 99% CI of

Why confidence intervals work? The mean of the sampling distribution equals the unknown population mean The standard error of the sampling distribution equals the value obtained from dividing the population standard deviation by the square root of the sample size The shape of the sampling distribution approximates a normal distribution when the central limit theorem requirement is met.

A more practical computation of confidence interval of  The (1- ) 100% confidence interval for a population mean  can be computed based on sample standard deviation s and t distribution which is known as following: (1- )100% CI for

Confidence Intervals for the Difference between Two Means (I) From the central limit theorem, it can be demonstrated that sampling distribution of X1-X2 is normally distributed with a mean of 1- 2 and a variance equal to 12/n1+ 22/n2. Its square root is the standard error of the difference between two means.

Confidence Intervals for the Difference between Two Means (II) A practical way of estimating standard error of the difference is using pooled standard deviation, Sp, using sample variances S12 and S22. Sp2 can be computed as following:

Confidence Intervals for the Difference between Two Means (III) 95% confidence interval for 1- 2: In practice, (1-)100 % CI for 1- 2:

Narrow Confidence Intervals Increasing the sampling size Increasing precision by reducing measurement (and other nonrandom) errors, thus producing a smaller variance Reducing the confidence level

Determination of Sample Size (I) “How much error can I live with in estimating the population mean?” or “What level of confidence is needed in the estimate?” (95% CI? or 99% CI?) “How much variability exists in the observation?” (s?)

Determination of Sample Size (II) Where: d is a measure of how close we need to come to the population mean  THINK: In practice, we won’t be able to know , how should above equation be revised?