Statistics for the Social Sciences Psychology 340 Spring 2010 Making estimations

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?

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. Two kinds of estimation Point estimates A single score Interval estimates A range of scores μ = ?

Estimation μ = ? Advantage Disadvantage Two kinds of estimation Point estimates Interval estimates Confidence of the estimate Little confidence of the estimate A single score A range of scores “the mean is 85” “the mean is somewhere between 81 and 89”

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)

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) Why the sample mean? 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.

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) Margin of error 1) A test statistic value (e.g., a z-score) 2) The standard error (the difference that you’d expect by chance)

Estimation Both kinds of estimates use the same basic procedure 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

Estimation Both kinds of estimates use the same basic procedure 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.

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

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? 95%

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? From the table: z(1.96) =.0250 So the confidence interval is: 83.04 to 86.96 or 85 ± 1.96 2.5% 95%

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? From the table: z(1.65) =.0500 So the confidence interval is: 83.35 to 86.65 or 85 ± 1.65 5% 90%

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? From the table: z(1.65) =.0500 So the confidence interval is: 80.88 to 89.13 or 85 ± 4.13 5% 90%

Estimation in other designs 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 Confidence interval How do we find this? Diff. Expected by chance

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 tcrit needed for a 95% confidence interval? 2.5% so two tails with 2.5% in each 2.5%+2.5% = 5% or α = 0.05, so look here 95% 95% in middle

Estimates with t-scores 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. What two critical t-scores do 95% of the data lie between? From the table: tcrit =+2.064 So the confidence interval is: 82.94 to 87.06 95% confidence or 85 ± 2.064 2.5% 95%

Estimation in other designs Estimating the difference between two population means based on two related samples Confidence interval Diff. Expected by chance

Estimation in other designs Estimating the difference between two population means based on two independent samples Confidence interval Diff. Expected by chance

Estimation Summary Design Estimation (Estimated) Standard error One sample, σ known One sample, σ unknown Two related samples, σ unknown Two independent samples, σ unknown

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?

Design Summary 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? Design One sample, σ known, 1 score per sub One sample z One sample, σ unknown, 1 score per One sample t 2 related samples, σ unknown, 1 score per - or – 1 sample, 2 scores per sub, σ unknown Related samples t Independent samples-t Two independent samples, σ unknown, 1 score per sub

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 Msmile = 7 and Mfrown = 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? Related samples t 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. Related samples CI 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? 1 sample z