Making estimations Statistics for the Social Sciences Psychology 340 Spring 2010

PSY 340 Statistics for the Social Sciences Statistical analysis follows design Are you looking for a difference between groups? Are you estimating the mean (or a mean difference)? Are you looking for a relationship between two variables?

PSY 340 Statistics for the Social Sciences Estimation So far we’ve been dealing with situations where we know the population mean. However, most of the time we don’t know it. μ = ? Two kinds of estimation –Point estimates A single score –Interval estimates A range of scores

PSY 340 Statistics for the Social Sciences Estimation μ = ? Two kinds of estimation –Point estimates –Interval estimates AdvantageDisadvantage A single score A range of scoresConfidence of the estimate Little confidence of the estimate “the mean is 85” “the mean is somewhere between 81 and 89”

PSY 340 Statistics for the Social Sciences Estimation Both kinds of estimates use the same basic procedure –The formula is a variation of the test statistic formula (so far we know the z-score)

PSY 340 Statistics for the Social Sciences Estimation Both kinds of estimates use the same basic procedure –The formula is a variation of the test statistic formula (so far we know the z-score) 1) It is often the only piece of evidence that we have, so it is our best guess. 2) Most sample means will be pretty close to the population mean, so we have a good chance that our sample mean is close. Why the sample mean?

PSY 340 Statistics for the Social Sciences Estimation Both kinds of estimates use the same basic procedure –The formula is a variation of the test statistic formula (so far we know the z-score) 1) A test statistic value (e.g., a z-score) 2) The standard error (the difference that you’d expect by chance) Margin of error

PSY 340 Statistics for the Social Sciences Estimation –Step 1: You begin by making a reasonable estimation of what the z (or t) value should be for your estimate. For a point estimation, you want what? z (or t) = 0, right in the middle For an interval, your values will depend on how confident you want to be in your estimate –What do I mean by “confident”? »90% confidence means that 90% of confidence interval estimates of this sample size will include the actual population mean Both kinds of estimates use the same basic procedure

PSY 340 Statistics for the Social Sciences Estimation –Step 1: You begin by making a reasonable estimation of what the z (or t) value should be for your estimate. For a point estimation, you want what? z (or t) = 0, right in the middle For an interval, your values will depend on how confident you want to be in your estimate –Step 2: You take your “reasonable” estimate for your test statistic, and put it into the formula and solve for the unknown population parameter. Both kinds of estimates use the same basic procedure

PSY 340 Statistics for the Social Sciences Estimates with z-scores Make a point estimate of the population mean given a sample with a X = 85, n = 25, and a population σ = 5. So the point estimate is the sample mean

PSY 340 Statistics for the Social Sciences Estimates with z-scores Make an interval estimate with 95% confidence of the population mean given a sample with a X = 85, n = 25, and a population σ = 5. 95% What two z-scores do 95% of the data lie between?

PSY 340 Statistics for the Social Sciences Estimates with z-scores Make an interval estimate with 95% confidence of the population mean given a sample with a X = 85, n = 25, and a population σ = 5. What two z-scores do 95% of the data lie between? So the confidence interval is: to From the table: z(1.96) = % 2.5% or 85 ± 1.96

PSY 340 Statistics for the Social Sciences Estimates with z-scores Make an interval estimate with 90% confidence of the population mean given a sample with a X = 85, n = 25, and a population σ = 5. What two z-scores do 90% of the data lie between? So the confidence interval is: to From the table: z(1.65) =.0500 or 85 ± % 90%

PSY 340 Statistics for the Social Sciences Estimates with z-scores Make an interval estimate with 90% confidence of the population mean given a sample with a X = 85, n = 4, and a population σ = 5. What two z-scores do 90% of the data lie between? So the confidence interval is: to From the table: z(1.65) =.0500 or 85 ± % 90%

PSY 340 Statistics for the Social Sciences Estimation in other designs Confidence interval Diff. Expected by chance Estimating the mean of the population from one sample, but we don’t know the σ How do we find this? Use the t-table

PSY 340 Statistics for the Social Sciences Estimates with t-scores Confidence intervals always involve + a margin of error This is similar to a two-tailed test, so in the t-table, always use the “proportion in two tails” heading, and select the α-level corresponding to (1 - Confidence level) What is the t crit needed for a 95% confidence interval? 95% 95% in middle 2.5% so two tails with 2.5% in each 2.5%+2.5% = 5% or α = 0.05, so look here

PSY 340 Statistics for the Social Sciences Make an interval estimate with 95% confidence of the population mean given a sample with a X = 85, n = 25, and a sample s = 5. Estimates with t-scores What two critical t- scores do 95% of the data lie between? So the confidence interval is: to From the table: t crit = % 2.5% or 85 ± % confidence

PSY 340 Statistics for the Social Sciences Estimation in other designs Confidence interval Diff. Expected by chance Estimating the difference between two population means based on two related samples

PSY 340 Statistics for the Social Sciences Estimation in other designs Confidence interval Estimating the difference between two population means based on two independent samples Diff. Expected by chance

PSY 340 Statistics for the Social Sciences Estimation Summary DesignEstimation (Estimated) Standard error One sample, σ known One sample, σ unknown Two related samples, σ unknown Two independent samples, σ unknown

PSY 340 Statistics for the Social Sciences Statistical analysis follows design Questions to answer: Are you looking for a difference, or estimating a mean? Do you know the pop. SD (σ)? How many samples of scores? How many scores per participant? If 2 groups of scores, are the groups independent or related? Are the predictions specific enough for a 1- tailed test?

PSY 340 Statistics for the Social Sciences Design Summary Design One sample, σ known, 1 score per sub One sample, σ unknown, 1 score per 2 related samples, σ unknown, 1 score per - or – 1 sample, 2 scores per sub, σ unknown Two independent samples, σ unknown, 1 score per sub Independent samples-t Related samples t One sample t One sample z Questions to answer: Are you looking for a difference, or estimating a mean? Do you know the pop. SD (σ)? How many samples of scores? How many scores per participant? If 2 groups of scores, are the groups independent or related? Are the predictions specific enough for a 1- tailed test?

PSY 340 Statistics for the Social Sciences Estimates with z-scores Researchers used a sample of n = 16 adults. Each person’s mood was rated while smiling and frowning. It was predicted that moods would be rated as more positive if smiling than frowning. Results showed M smile = 7 and M frown = 4.5. Are the groups different? Questions to answer: Are you looking for a difference, or estimating a mean? Do you know the pop. SD (σ)? How many samples of scores? How many scores per participant? If 2 groups of scores, are the groups independent or related? Are the predictions specific enough for a 1- tailed test? Researcher measures reaction time for n = 36 participants. Each is then given a medicine and reaction time is measured again. For this sample, the average difference was 24 ms, with a SD of 8. With 95% confidence estimate the population mean difference. A teacher is evaluating the effectiveness of a new way of presenting material to students. A sample of 16 students is presented the material in the new way and are then given a test on that material, they have a mean of 87. How do they compare to past classes with a mean of 82 and SD = 3? Related samples t Related samples CI 1 sample z