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Psych 2220, one-sample t test Psych 2220

Marketplace > Ohio State University > Psych 2220 > Psych 2220 one sample t test
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Here is an intro of notes to one-sample t tests, I will upload some more notes with practice problems worked through tomorrow :)
COURSE
Data Analysis in Psychology
PROF.
Joseph Roberts
TYPE
Class Notes
PAGES
4
WORDS
CONCEPTS
Stats, psych, Data Analysis, OSU, ohio state, neuroscience, 2220, hitchcock, college, help, Homework, Math, t test, one sample
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This 4 page Class Notes was uploaded by MadsSwart on Saturday March 5, 2016. The Class Notes belongs to Psych 2220 at Ohio State University taught by Joseph Roberts in Winter 2016. Since its upload, it has received 14 views.

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Date Created: 03/05/16
Chapter 9 Notes: One-Sample T Tests - Quick review o normally shaped distribution of scores with a specific µ and σ– convert to z scores for the probability of randomly sampling specific values o population with specific µ and σ—convert sample mean, M into z statistic- for the probability of obtaining that mean by random sampling  We no longer need to know that the original distribution of scores is normal (if N >30) - What’s New o Sample N>30, and specific µ, convert sample mean, M into standardized test statistic for the probability of randomly sampling specific values  We don’t know σ so this must be estimated  Use samples standard deviation, s, as our best point estimate for σ  Can similarly adjust confidence interval and effect size formulas o If population has specific µ, converting sample mean M into t statistic will tell us about the probability of obtaining that mean by random sampling.  We no longer need to know that the original distribution of scores is normal (if N>30)  We no longer need to know the population’s σ value o When we discard assumptions about the population we sampled from, we can better accommodate the real world (which deviates from those assumptions)  The one sample t statistic is exactly like the z statistic, but with one less assumption, and a different table to lookup values in. M−μ M M−μ M M=¿ = σ M σ √N Z ¿ M−μ M−μ t = M = M M M s √ N - One sample t test: o A test statistic that expresses the distance between a sample mean M and some suspected population mean µ, scaled to an inferentially valuable indication of variability: standard error of the mMan (s )  Look and realize that this formula is already one that we know- M−μ M−μ t = M = M M sM s √ N o The hypothesis test we perform to determine whether we should reject a null hypothesis about µ  Follows the same approach as the z test (approximately)  H 1 µ ≠ 50 (for any specific focal value of µ)  H 0 µ = 50 (Formally, µ -50=0) (look for the zero-point) o Follows the same approach as the z test-but we need to know the sample size N o Degrees of freedom (df) for a one-sample t test: N – 1 o There are actually many distributions for the t statistic, each corresponding to a different sample size. So we use df to isolate the specific distribution –and corresponding critical values for our hypothesis test. o Table B-2: Critical Values in the t distributions  Recall that there’s one standard normal curve –z distribution- but we have lots of similar-yet-different t distributions  So the table only shows critical values for certain levels of significance (.10, .05 and.01) o Notice how the denominator can be represented in terms of sample 2 variance (s ) rather than sample standard deviation (s): M−μ M−μ tM= M = M sM s N √ 2 o That hidden variance term (s ) will become important again later, so keep track of how to find it. - So far o We discovered that one-sample t tests are almost exactly like z tests, but used when we don’t know the population’s σ o This better accommodates real world use of statistics  Do psychology majors’ GPAs differ from the OSU average?  Are students favorable toward the new OSU president? (significantly more positive than a hypothesized population average that reflects ‘neutral’ or ‘no opinion’) - Where we are o We know something about normal distributions. If we’re sampling from a normal distribution, great-if not, we should try to get N>30 in our sample. The central limit theorem tells us that a distribution of means would then be normally shaped.  Then:  TEST HYPOTHESIS (compare sample mean M to some population µ)  CONFIDENCE INTERVAL (for µ, centered on M)  EFFECT SIZE (of a difference between M and µ) - The problem of variance o We use the t distribution if we don’t have σ  Different t distributions for different sample sizes o Z distribution doesn’t vary with sample size, because it’s built from know population properties o t distribution varies with sample, size, because we use our observed sample to infer population properties. With increasing sample size, our inference is more accurate  N  ∞, t distribution  z distribution - Factors affecting significance of t tests o Unstandardized EFFECT SIZE o Magnitude of measured STANDARD DEVIATION o SAMPLE SIZE o ONE VS TWO-TAILED TEST - What we can do o Test for difference between one score and a population  Convert to z o Test for differences between sample and population with known µ and σ  Use z-statistics, attending to sample sizeMZ o Test for differences between sample and population with unknown σ  Use sample s as a best-guess substitute for σ  Use t-statistics, attending to sample size: t & df Excercise 1: Average recent monthly profits (M) vs. long-run monthly average profit (µ=\$5,000) a. H1: profit is too much lower than average, µ < 5,000 b. H0: profit is NOT too much lower than average, µ ≥ 5,000 2. Data: the past 3 months profits were \$4500, \$3100, \$3900 3. Sample mean M & standard deviation s (what is N?) a. M= 3833 b. S= 702 c. N= 3 4. Standard error of the mean, sM σ = σ a. SM= (702/ √3) = 405 M √N  s = s M √N 5. One-sample t statistic t = M−μ M = M−μ M M s s a. t=(3833 – 5000) / 405 = -2.88 M √ N 6. Compare to critical value in t table for (df = N = 1) a. tcritical2 for df = 2 & one-tailed α = .05 7. Reject null? a. -2.88 is NOT more extreme than -2.92. so we fail to rejec0 H b. Profits do not seem significantly lower than the average, based on this sample. 8. Effect Size a. Cohen’s d= (3833 – 5000) / 702 i. d= -1.66 Cohen sd= (M−μ) σ b. THESE three months’ profits were 1.66 standard deviations below the long-run average (descriptively speaking) c. Effect size alone doesn’t tell us whether all months’ profits are systematically lower than average. 9. 90% Confidence Interval M lower samplecrit M M upper samplecrit M a. µupper 3833 + 2.92 * (405) = 5015 b. µlower 3833 – 2.92 * (405) = 2650 c. the 90% confidence interval includes \$5,000

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