# Statistics for the Behavioral and Social Sciences: A Brief ... Hypothesis Tests with Means of Samples Chapter 6 Copyright 2011 by Pearson Education, Inc. All rights reserved Chapter Outline The Distribution of Means Hypothesis Testing with a Distribution of Means: The Z Test Hypothesis Tests about Means of Samples (Z Tests) in Research Articles

Advanced Topic: Estimation and Confidence Intervals Advanced Topic: Confidence Intervals in Research Articles Estimation & Confidence Intervals Point Estimates: Estimation of the pop. Mean () Estimating based on the scores in a sample is an important approach in experimental and survey research. When is unknown, its best is the sample mean. The accuracy of the population mean

estimate is SDM (standard error of the mean). Estimation & Confidence Intervals Confidence Interval used to get a sense of the accuracy of an estimated population mean Estimates a RANGE of all possible means likely to include the true POP. MEAN. It is the range of population means from which it is not highly unlikely that you could have obtained your sample mean.

(e.g., Range means for todays temp. 70F to 90F) Estimation & Confidence Intervals Want to be 95% confident a particular distribution includes the pop. mean? Estimate pop. mean based on N = 64, Mean = 220, & SDM = 6 What is the (confident) lower and the upper limit? Temperature 70F - 90F LOWER UPPER

Estimation & Confidence Intervals Want p < .05, distribute p on both extremes, p < .025 = Z-score of 1.96 Determine distribution: Sample M = 220, SDM = 6 Estimated Pop. Mean = 220 Lower limit: X = (SDM)(-Z) + M = (6)(-1.96)+ 220 = 208.24 Upper limit: X = (SDM)(Z)+ M = (6)(1.96)+ 220 = 231.76 Pop. mean is between 208.24 and 231.76. 95% Confidence Interval

-1.96 208.24 Population Mean? +1.96 231.46 Interval Estimation

Margin of Error (ME) = (ZCritical p-Value)(SDM) ME = Width of confidence interval Estimate pop. mean based on N = 64 Mean = 220, & SDM = 6 Using a 95% Confidence: ME = (1.96)(6) = 11.76 + 220 Estimated CI = 208.24, 231.76 ALSO: ME = 231.76-208.24 = 23.52/2 =11.76 Center of CI= sample statistic (sample mean/proportion) Interval Estimation How about with a p < .01 (or 99% confident interval);

Then p < .005 for both extremes Z of .5% = 2.57 = (6)(2.57)+ 220 = 204.58 and 235.42 **99% chance the pop. mean will fall within this interval **1% chance ppp. pppp will not be in this 99% range. 99% Confidence Interval -2.57 204.58

Population Mean? +2.57 235.42 Interval Estimation Unoccupied seats on flights cause the airlines to lose revenue. Suppose a large airline wants to estimate its average number of occupied seats per flight over the

past year. To accomplish this, the records of 225 flights are randomly selected from the files, and the number of unoccupied seats is noted for each of the sampled flights. The sampling mean M = 11.6 seats & SDM = 4.1 Estimate the Population Mean, the mean number of unoccupied seats per flight during the past year, using a 95% confidence interval. Can we estimate the population mean of unoccupied seats per flight?

P < .05 you want to estimate both extremes (2-tailed). .05/2 = .025 = Z = 1.96 Lower limit X = (SDM)(-Z)+ M (4.1)(-1.96)+ 11.6 = 3.6 Upper limit X = (SDM)(Z)+ M (4.1)(1.96)+ 11.6 = 19.6 We are 95% confident that the population mean of unoccupied seats will fall between 3.6 and 19.6 If we used P < .01? Than P < .005 Z = 2.57 = (4.1)(2.57)+ 11.6 = 1.1 and 22.137

Confidence Intervals In Research Articles Confidence intervals are becoming more common in research articles in some fields. Key Points A distribution of means is the comparison distribution when studying a sample of more than one individual. The distribution of means has the same mean as the corresponding

population of individuals. The variance of the distribution of means is the variance of the population of individuals divided by the number of individuals in each sample. Its standard deviation is the square root of its variance. The shape of a distribution of means is close to normal if the number of participants in the samples is at least 30 or if the population of individuals follows a normal curve. Z tests are hypothesis tests with a single sample of more than one individual and a known population. The comparison distribution for Z tests is a distribution of means. Z tests are rarely found in research articles.

Key Points for Advanced Topic: Confidence Intervals The sample mean is the best estimate for the population mean when the population mean is unknown. The accuracy of the

estimate is the standard deviation of the distribution of means. Confidence intervals are ranges of possible means that are likely to include the population mean. For a 95% confidence interval, the Z score range is -1.96 to +1.96. For a 99% confidence interval, the range is from -2.58 standard deviations to +2.58 standard deviations. Confidence intervals are sometimes reported in research articles.