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Reasoning in Psychology Using Statistics Psychology 138 2013 • Describe the typical college student μ= ? – If we can’t measure the entire population of students, how do we get the population means for these variables? Age μ= ? hours of studying per week μ= ? μ=? hours of sleep per night Reasoning in Psychology Using Statistics pizza consumption – If we can’t measure the entire population of students, how do we get the population means for these variables? μ= ? • Estimate it based on what we do know – On information from a sample • Two kinds of estimation – Point estimates • A single score – Interval estimates • A range of scores X Estimation Reasoning in Psychology Using Statistics • Describe the typical college student – Point estimates – Interval estimates “12 hrs” “2 to 21 hrs” “19 yrs” “17 to 21 yrs” Age “8 hrs” “4 to 10 hrs” hours of sleep per night Reasoning in Psychology Using Statistics hours of studying per week “1 per wk” “0 to 8 per wk” pizza consumption Estimate the number of people attending lecture today How confident are you that your estimate is correct? “Not real confident, maybe 20%” “50 students” “Somewhere between 20 and 80 students” Estimation Reasoning in Psychology Using Statistics “Fairly confident, maybe 90%” Kinds of estimation Point estimate “50 students” Interval estimate “Somewhere between 20 and 80 students” Estimation Reasoning in Psychology Using Statistics Advantage Disadvantage A single score Little confidence in the estimate Confidence in the estimate A range of scores • Both kinds of estimates use the same basic procedure – The formula is a variation of the test statistic formulas (we’ll start with the z-score) mX = X ± zX (s X ) Do some adding/subtracting zX (s X ) = X - mX Multiply both sides by sX zX = Estimation Reasoning in Psychology Using Statistics X - mX sX This is what we want to estimate • Both kinds of estimates use the same basic procedure – The formula is a variation of the test statistic formulas (we’ll start with the z-score) mX = X ± zX (s X ) 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 Reasoning in Psychology Using Statistics • Both kinds of estimates use the same basic procedure – The formula is a variation of the test statistic formulas (we’ll start with the z-score) mX = X ± zX (s X ) Margin of error 1) 2) A test statistic value (e.g., a z-score) based on design (z or t) and level of confidence The standard error (the difference that you’d expect by chance) Based on sample size (n) and population standard deviation (σ) Estimation Reasoning in Psychology Using Statistics • Finding the right test statistic for your estimate mX = X ± zX (s X ) – 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 mX μ= ? zX = Raw scores X - mX sX transform z scores Z=0 Estimation Reasoning in Psychology Using Statistics • Finding the right test statistic (z or t) mX = X ± zX (s X ) – 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 Actual population mean – What do I mean by “confident”? μ » 90% confidence means that 90% of confidence interval estimates of this sample size will include the actual population mean 9 out of 10 intervals contain μ Estimation Reasoning in Psychology Using Statistics • Finding the right test statistic (z or t) mX = X ± zX (s X ) – 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 – Computing the point estimate or the confidence interval: • Step 1: Take your “reasonable” estimate for your test statistic • Step 2: Put it into the formula • Step 3: Solve for the unknown population parameter Estimation Reasoning in Psychology Using Statistics Make a point estimate of the population mean given a sample with a X = 85, n = 25, and a population σ = 5. æ 5 ö mX = X ± zX (s X ) = 85 ± (0)ç ÷ = 85 è 25 ø sample mean serves as the center So the point estimate is the sample mean z (or t) = 0, right in the middle Estimates with z-scores Reasoning in Psychology Using Statistics sX = s n 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. mX = X ± zX (s X ) What two z-scores do 95% of the data lie between? 95% Estimates with z-scores Reasoning in Psychology Using Statistics 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. æ 5 ö = 86.96 What two z-scores do 95% of the data mX = X ± zX (s X ) = 85 ± (1.96)ç ÷ è 25 ø= 83.04 lie between? So the 95% confidence interval is: 83.04 to 86.96 or 85 ± 1.96 z : 1.8 1.9 2.0 : 0.02 : .0344 .0274 .0217 : 0.03 : .0336 .0268 .0212 : 0.04 : .0329 .0262 .0207 : 0.05 : .0322 .0256 .0202 : 0.06 : .0314 .0250 .0197 : From the table: z(1.96) =.0250 2.5% 2.5% 95% Estimates with z-scores Reasoning in Psychology Using Statistics 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. æ 5 ö = 86.65 What two z-scores do 90% of the data mX = X ± zX (s X ) = 85 ± (1.65)ç ÷ è 25 ø= 83.35 lie between? So the 90% confidence interval is: 83.35 to 86.65 or 85 ± 1.65 From the table: z(1.65) =.0500 5% 5% 90% Estimates with z-scores Reasoning in Psychology Using Statistics 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. æ 5 ö = 89.13 What two z-scores do 90% of the data mX = X ± zX (s X ) = 85 ± (1.65)ç ÷ è 4 ø = 80.88 lie between? So the confidence interval is: 80.88 to 89.13 or 85 ± 4.13 From the table: z(1.65) =.0500 5% 5% 90% Estimates with z-scores Reasoning in Psychology Using Statistics • The size of the margin of error related to: mX = X ± zX (s X ) – Sample size • As n increases, the margin of error gets narrower (changes the standard error) – Level of confidence • As confidence increases (e.g., 90%-> 95%), the margin of error gets wider (changes the critical test statistic values) Estimation Reasoning in Psychology Using Statistics • Two kinds of estimates that use the same basic procedure – The formula is a variation of the test statistic formulas Different Designs: Estimating the mean of the population from one or two samples, but we don’t know the σ m X = X ± zX (s X ) mX = [sample mean(s)] ± (tcrit )(estimated standard error ) Center/point estimate? How do we find How do we find this? this? Depends on the design Use the t-table & your Depends on the design (what is being estimated) confidence level sX D X XA - XB sD Estimation in other designs Reasoning in Psychology Using Statistics sXA - XB 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? so two tails with 2.5%+2.5% = 5% or 95% in middle 2.5% in each α = 0.05, so look here 0.10 df : 5 6 : 0.20 : 1,476 1.440 : Proportion in one tail 0.05 0.025 Proportion in two tails 0.10 0.05 : : 2.015 2.571 1.943 2.447 : : 0.01 0.005 0.02 : 3.365 3.143 : 0.01 : 4.032 3.707 : 2.5% Estimates with t-scores Reasoning in Psychology Using Statistics 2.5% 95% Estimating the difference between the population mean and the sample mean based when the population standard deviation is not known Confidence interval Diff. Expected by chance m X = X ± (tcrit )(sX ) s sX = n Estimation in other designs Reasoning in Psychology Using Statistics 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. two critical tæ 5 ö = 87.06 What scores do 95% of the m X = X ± tcrit (sX )= 85 ± (2.064) ç ÷ è 25 ø = 82.94 data lie between? df = n - 1 = 25 - 1 = 24 From the table: tcrit =+2.064 0.10 df : 24 25 : 0.20 : 1.318 1.316 : Proportion in one tail 0.05 0.025 0.01 Proportion in two tails 0.10 0.05 0.02 : : : 1.711 2.064 2.492 1.708 2.060 2.485 : : : So the confidence interval is: 82.94 to 87.06 or 85 ± 2.064 0.005 0.01 : 2.797 2.787 : 2.5% 2.5% 95% Estimation in one sample t-design Reasoning in Psychology Using Statistics Estimating the difference between two population means based on two related samples Confidence interval mD = D ± (t crit )(sD ) Diff. Expected by chance sD = sD nD Estimation in related samples design Reasoning in Psychology Using Statistics • Dr. S. Beach reported on the effectiveness of cognitive-behavioral therapy as a treatment for anorexia. He examined 12 patients, weighing each of them before and after the treatment. Estimate the average population weight gain for those undergoing the treatment with 90% confidence. Differences (post treatment - pre treatment weights): 10, 6, 3, 23, 18, 17, 0, 4, 21, 10, -2, 10 Related samples estimation mD = D ± (t crit )(sD ) sD = D = (XA - XB ) Confidence level 90% dfD = nD - 1 = 11 D D=å tcrit = ±1.796 nD sD = mD = 10 ± (1.796)(2.38) CI(90%)= 5.72 to 14.28 SSD nD - 1 SSD = å ( D - D ) Estimation in related samples design Reasoning in Psychology Using Statistics sD nD 2 Estimating the difference between two population means based on two independent samples Confidence interval mA - mB = (X A - X B ) ± (tcrit )(sX Diff. Expected by chance sXA - XB A -X B ) sP2 sP2 = + nA nB Estimation in independent samples design Reasoning in Psychology Using Statistics • Dr. Mnemonic develops a new treatment for patients with a memory disorder. He randomly assigns 8 patients to one of two samples. He then gives one sample (A) the new treatment but not the other (B) and then tests both groups with a memory test. Estimate the population difference between the two groups with 95% confidence. X A = 44.5 X B = 50 sA = 7.19 sB = 9.13 Independent samples t-test situation mA - mB = (X A - X B ) ± (tcrit )(sX Confidence level 95% df = nA + nB - 2 = 6 tcrit = ±2.45 mA - mB = 5.5 ± (2.45)(5.81) CI(95%)= -8.73to 19.73 s XA - XB A -X B ) sP2 sP2 = + = 5.81 nA nB s 2p s df ) + ( s df ) ( = 2 A A 2 B B dfA + dfB Estimation in independent samples design Reasoning in Psychology Using Statistics • Notice that this interval includes zero -8.73 19.73 0 • If we had instead done a hypothesis test with an α = 0.05, what would you expect our conclusion to be? H0: “there is no difference between the groups” - Fail to reject the H0 CI(95%)= -8.73to 19.73 Relating estimates to hypothesis tests Reasoning in Psychology Using Statistics Design Estimation One sample, σ known mX = X ± zX (s X ) One sample, σ unknown m X = X ± (tcrit )(sX ) Two related samples, σ unknown mD = D ± (t crit )(sD ) Two independent samples, σ unknown mA - mB = (X A - X B ) ± (tcrit )(sX (Estimated) Standard error sX = n s sX = n sD sD = nD ) A -X B Estimation Summary Reasoning in Psychology Using Statistics s sXA - XB sP2 sP2 = + nA nB • Practice computing and interpreting confidence intervals In labs Reasoning in Psychology Using Statistics