§ Sampling Distributions and Confidence Intervals §

Lesson 1: Introduction

In prior lessons, you have been concerned with the distributions of raw data, x values. In this lesson, the focus will switch the distribution of sample averages, a sampling distribution. If one starts with raw data in a population of x values (e.g. a normal distribution of x values) and obtains a sample from that population, then the average of those x values results in xbar, the sample mean.  The best point estimate of the population mean (m) is xbar.  A confidence interval gives a range of possible values for m at a given level of confidence.

§ Sampling Distribution §

The CLT states that the distribution of xbars taken from a population with mean, m , and standard deviation, s , will be normally distributed with a mean of the xbars equal to m _xbar = m and a standard deviation (standard error), s _xbar= s /Ö (n). The CLT applies to the distribution of xbars from any population as long as the sample size is large (n > 30) and to the distribution of xbars from an normal population at any sample size. The distribution of xbars is not as wide as the original distribution of xs. (click me)

Since the population mean, m , is usually unknown, an estimate of it is desirable. The sample mean, xbar, is used as a point estimate of m . But, a single xbar is very unlikely to equal m . Thus, a range of possible values for m is used with the point estimate, xbar, in the middle of the range. The amount added to xbar and subtracted from xbar is the maximum error. The resulting range is xbar + maximum error = upper limit of the confidence interval; and, xbar - maximum error = lower limit of the confidence interval. The maximum error is made up of two parts. First, a Z score (or t score) counts the number of standard errors one is willing to go on each side xbar; and second, the size of the standard error, s _xbar= s /Ö (n) must be calculated. These two are multiplied to give the maximum error. A confidence interval is two maximum errors wide with xbar in the middle. In selecting a Z score or t score one must consider sample size, n (and the resulting degrees of freedom = n -1) and the source of the standard deviation.   (click me)

Source of Standard Deviation

In addition, one must select an alpha value, a, in order to select a Z score or t score. Alpha is the proportion of the time one is willing to not capture the true population mean, m (a parameter), with the confidence interval constructed around the sample mean, xbar (a statistic). The usual values for alpha are 0.01, 0.05 and 0.10. (click me)

When the decision has been made to use a Z score, Z_a/2, one must look up 0.5 - a/2 = probability in the body of the Z table. Alpha is divided by two, since there are two ends of the confidence interval (the upper limit and the lower limit). Pick the closest probability, and then read the Z score from the edges of the table. (cick me)

When the decision has been made to use a t score, t_a/2,df, one must simply go to the t table and find a/2 at the top of a column and the degrees of freedom, n -1, along the left side of the table. The t score is found at the intersection of that column and row. When the degrees of freedom is greater than 30, we will used the convention of skipping to the infinity degrees of freedom row. That t score at infinity degrees of freedom is equal to a Z score utilizing the same a/2, t_a/2,infinity = Z_{a/2. (click me)}

Go on to Lesson 2: Examples
or
Go back to Sampling Distributions and Confidence Intervals

Please reference "BA501 (your last name) Assignment name and number" in the subject line of either below.

Back to top