PSYC2009 Study Guide - Final Guide: Confidence Interval, Halfwidth And Fullwidth Forms, Sampling Distribution
Introduction to Confidence Intervals
• Confidence interval - is a range of values for a statistic that contains a specified percentage of
the sampling distribution of that statistic.
• Confidence level - the specified percentage.
Confidence Interval for the Mean
• Consists of a range of values starting somewhere below the sample mean and ending the
same distance above it, eg: 95% confidence interval.
• The range of values includes "plausible" values for the population mean.
• Lower limit is the same distance from the mean as the upper limit.
• The half-width of a confidence interval is the distance from the mean to either of the limits.
• Can also express the confidence interval as eg: 53 ± 4.8.
• The 95% refers to how confident we are that the population actually lies between two values.
• We can't be 100% confident because that would produce an infinitely wide confidence
interval.
How to calculate CI for the mean
• We can think of the sample mean as consisting of the population mean + sampling error: X=μ +
e
• Note that the error (e) may be positive or negative.
• The standard error of the sample mean is: sx=s/√N, where s is the sample SD and N is the no.
of observations in the sample.
• The SE is one component of the half-width w, so the larger the SE, the bigger w is (the bigger
the CI is).
• t statistic - the ratio of the sample mean to its SE has a t distribution as a sampling
distribution.
• t distribution – sampling distribution of a mean
• By rearranging the CI statement, we can see that if we knew the sampling distribution of the
ratio of the difference between the sample mean and population mean to the SE, then we
could find the t statistic.
• i.e. We start with a statement of what our 95% CI Means: Pr (x̅ − w < μ < x̅ + w)= .95,
• We subtract the sample mean from all three terms to get: Pr (−w < μ - x̅ < w) = Pr (−t*s < μ − x̅
< t*s)= .95,
• And then divide all three terms by the SE to get: Pr (−t < (μ − x̅)/s < t) = .95.
• The t distribution is quite similar to a normal distribution but has larger tails, which means that
Cis based on the t distribution will be wider than they would be if they were based on the
normal distribution.
Steps in computing a 95% CI for the mean
1. Collect a sample of N observations.
2. Calculate the sample mean, SD and SE.
3. Ascertain your desired confidence level (95%). Convert the confidence level to a proportion
(0.95) and subtract it from 1 to get the proportion of the sample distribution to be excluded
from the confidence interval (α = 0.05).
4. Using Table A.3, find the cell entry in the row corresponding to df = N - 1. Then the half-width:
w = t/SE.
5. The resulting CI statement is Pr (x̅ − w < μ < x̅ + w)= 1 - α = .95.
Properties of Confidence Intervals
1. The locations and widths of confidence intervals vary solely due to sampling error.
2. If we repeated this sampling procedure a very large number of times, the percentage of
intervals containing the true mean would home in on 95%, Technically speaking, the
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Document Summary
Introduction to confidence intervals: confidence interval - is a range of values for a statistic that contains a specified percentage of the sampling distribution of that statistic, confidence level - the specified percentage. Confidence interval for the mean: consists of a range of values starting somewhere below the sample mean and ending the same distance above it, eg: 95% confidence interval. Cis based on the t distribution will be wider than they would be if they were based on the normal distribution. Steps in computing a 95% ci for the mean: collect a sample of n observations, calculate the sample mean, sd and se, ascertain your desired confidence level (95%). Then the half-width: w = t/se: the resulting ci statement is pr (x w < < x + w)= 1 - = . 95. Properties of confidence intervals: the locations and widths of confidence intervals vary solely due to sampling error.
