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Chapter 9: Introduction to the t statistic The t Statistic • The t statistic allows researchers to use sample data to test hypotheses about an unknown population mean. • The particular advantage of the t statistic is that the t statistic does not require any knowledge of the population standard deviation, i.e. σ. • When σ is unknown, we cannot use z score to test the hypothesis, but we can use t statistic instead. The t Statistic (cont’d.) • Thus, the t statistic can be used to test hypotheses about a completely unknown population; that is, both μ and σ are unknown, and the only available information about the population comes from the sample. • All that is required for a hypothesis test with t is a sample and a reasonable hypothesis about the population mean. The Estimated Standard Error and the t Statistic • The goal for a hypothesis test is to evaluate the significance of the observed discrepancy between a sample mean and the population mean. (M – μ) • Whenever a sample is obtained from a population you expect to find some discrepancy or "error" between the sample mean and the population mean. • This general phenomenon is known as sampling error. The Estimated Standard Error and the t Statistic (cont’d.) • The hypothesis test attempts to decide between the following two alternatives: 1. Is it reasonable that the discrepancy between M and μ is simply due to sampling error and not the result of a treatment effect? 2. Is the discrepancy between M and μ more than would be expected by sampling error alone? That is, is the sample mean significantly different from the population mean? The Estimated Standard Error and the t Statistic (cont.) • The critical first step for the t statistic hypothesis test is to calculate exactly how much difference between M and μ is reasonable to expect. • However, because the population standard deviation is unknown, it is impossible to compute the standard error of M as we did with z-scores in Chapter 8. • Therefore, the t statistic requires that you use the sample data to compute an estimated standard error of M. i.e. sM The Estimated Standard Error and the t Statistic (cont’d.) • This calculation defines standard error exactly as it was defined in Chapters 7 and 8, but now we must use the sample variance, s2, in place of the unknown population variance, σ2 (or use sample standard deviation, s, in place of the unknown population standard deviation, σ). • The resulting formula for estimated standard error is: s2 s sM = ── or sM = ── n n The Estimated Standard Error and the t Statistic (cont’d.) • The t statistic (like the z-score) forms a ratio. • The top of the ratio contains the obtained difference between the sample mean and the hypothesized population mean. • The bottom of the ratio is the standard error which measures how much difference is expected by chance. obtained difference Mμ t = ───────────── = ───── standard error sM The Estimated Standard Error and the t Statistic (cont’d.) • A large value for t (a large ratio) indicates that the obtained difference between the data and the hypothesis is greater than would be expected if the treatment has no effect. • For a small sample and unknown σ: • t ↑ more likely to be effective, more likely to reject H0 Degrees of Freedom • Degrees of freedom is the number of values which are involved in the final calculation of a statistic that is expected to vary, or free to vary. • In other words, these are the independent part of data used in calculation. It is used to know the accuracy of the sample population used in research. The larger the degree of freedom, larger the possibility of the entire population to be sampled accurately. Degrees of Freedom and the t Statistic • You can think of the t statistic as an "estimated z-score." • The estimation comes from the fact that we are using the sample variance s to estimate the unknown population variance σ. • With a large sample, the estimation is very good and the t statistic will be very similar to a zscore. • With small samples, however, the t statistic will provide a relatively poor estimate of z. Degrees of Freedom and the t Distribution • The value of degrees of freedom, df = n - 1, is used to describe how well the t statistic represents a z-score. • Also, the value of df will determine how well the distribution of t approximates a normal distribution. • For large values of df, the t distribution will be nearly normal, but with small values for df, the t distribution will be flatter and more spread out than a normal distribution. • df ↑ , then t ~ Normal z vs t • σ : the same for every sample z M M-μ : varies n • s : varies for different sample M t s n M-μ : varies s : varies Using the t-Distribution: σ Unknown It is, like the z distribution, a continuous distribution. It is, like the z distribution, bell-shaped and symmetrical. There is not one t distribution, but rather a family of t distributions. All t distributions have a mean of 0, but their standard deviations differ according to the sample size, n. The t distribution is more spread out and flatter at the center than the standard normal distribution As the sample size n increases, however, the t distribution approaches the standard normal distribution. Comparing the z and t Distributions When n is Small, 95% Confidence Level, n = 5 William Sealy Gosset Density of the t-distribution (red) for 1, 2, 3, degrees of freedom compared to the standard normal distribution (blue). Previous plots shown in green. Using the Student’s t-Distribution Table p. 290 2. a. s 2 b. sM SS n 1 S n = 288/8 = 36 63 2 4. df = n-1 = 20 n = 21 5. df = 15, check p. 703 a. top 5% t = 1.753 b. middle 95% t = 2.131 c. middle 99% t = 2.947 Hypothesis Setups for Testing a Mean () 10-* Degrees of Freedom and the t Distribution (cont’d.) • To evaluate the t statistic from a hypothesis test, 1.select an α level, 2.find the value of df for the t statistic, and 3.consult the t distribution table. – If the obtained t statistic is larger than the critical value from the table, you can reject the null hypothesis. – In this case, you have demonstrated that the obtained difference between the data and the hypothesis (numerator of the ratio) is significantly larger than the difference that would be expected if there was no treatment effect (the standard error in the denominator, the “random error”). Hypothesis Tests with the t Statistic • The hypothesis test with a t statistic follows the same four-step procedure that was used with zscore tests: 1. State the hypotheses and select a value for α. (Note: The null hypothesis always states a specific value for μ.) 2. Locate the critical region. (Note: You must find the value for df and use the t distribution table.) 3. Calculate the test statistic. t statistic 4. Make a decision. (Either "reject" or "fail to reject" the null hypothesis.) Example 9.1 (p. 292) n = 9, M = 13, SS = 72 : unknown population 1. H0: μ = 10 seconds (no preference/indifferent) H1: μ ≠ 10 seconds (not indifferent) α = 0.05 2. df = n-1 = 8 critical region: |t| > 2.306 3. s2 = SS/df = 72/8 = 9 s = 3 sM = s/√n = 1 t = (M- μ)/sM = 3 > 2.306 4. reject H0, i.e. it is more likely that μ ≠ 10 • Infants do prefer an attractive face, why? Hypothesis Tests with the t Statistic (cont’d.) • There are two general situations where this type of hypothesis test is used: 1. To determine the effect of treatment on a population mean μ 2. In situations where the population mean μ is unknown Hypothesis Tests with the t Statistic (cont’d.): case 1 (before/after) 1. In order to determine the effect of treatment on a population mean, you must know the value of μ for the original, untreated population. A sample is obtained from the population and the treatment is administered to the sample. If the resulting sample mean is significantly different from the original population mean, you can conclude that the treatment has a significant effect. Hypothesis Tests with the t Statistic (cont’d.): case 2 (theory/logic/guess) 2. Occasionally a theory or other prediction will provide a hypothesized value for an unknown population mean. A sample is then obtained from the population and the t statistic is used to compare the actual sample mean with the hypothesized population mean. A significant difference indicates that the hypothesized value for μ should be rejected. Hypothesis Tests with the t Statistic (cont’d.): Assumptions • Two basic assumptions are necessary for hypothesis tests with the t statistic: – The values in the sample must consist of independent observations. (random sample, iid) – The population that is sampled must be normal. Hypothesis Tests with the t Statistic (cont’d.) • Both the sample size n and the sample variance s influence the outcome of a hypothesis test. • The sample size is inversely related to the estimated standard error. Therefore, a large sample size increases the likelihood of a significant test. n↑ sM ↓ t ↑ • The sample variance, on the other hand, is directly related to the estimated standard error. Therefore, a large variance decreases the likelihood of a significant test. s↑ sM ↑ t ↓ p. 295 • μ = 40, n = 4, after treatment: M = 44, s2 = 16 a. two-tailed, α = 0.05, significant? sM = 4/2 = 2, df = 3, critical region: |t|>3.182 t = (44-40)/2 = 2 < 3.182 failed to reject H0 no significant effect after treatment b. n = 16, other things being equal, significant? sM = 4/4 = 1, df = 15, critical region: |t|>2.131 t = (44-40)/1 = 4 > 2.131 reject H0 does have a significant effect Measuring Effect Size for the t Statistic • Because the significance of a treatment effect is determined partially by the size of the effect and partially by the size of the sample, you cannot assume that a significant effect is also a large effect. • Therefore, it is recommended that a measure of effect size be computed along with the hypothesis test. Measuring Effect Size for the t Statistic (cont’d.) • For the t test, it is possible to compute an estimate of Cohen’s d just as we did for the zscore test in Chapter 8. The only change is that we now use the sample standard deviation instead of the population value (which is unknown). mean difference M μ estimated Cohen’s d = ─────────── = ────── standard deviation s Example 9.2 (p. 296) n = 9, M = 13, SS = 72 : unknown population H0: μ = 10 (indifferent) s2 = SS/df = 72/8 = 9 s = 3 Cohen’s d = (M- μ)/s = (13-10)/3 = 1 the size of treatment effect is equivalent to 1 standard deviation (shift to the right from μ = 10) Measuring Effect Size for the t Statistic (cont’d.) • As before, Cohen’s d measures the size of the treatment effect in terms of the standard deviation. • With a t test, it is also possible to measure effect size by computing the percentage of variance accounted for by the treatment. • This measure is based on the idea that the treatment causes the scores to change, which contributes to the observed variability in the data. Measuring the percentage of variance explained • r2 = variability accounted for (by treatment effect) / total variability = (SS1-SS2)/SS1 SS1:deviation from the hypothesized μ = total deviation = Σ(X-μ)2 SS2: deviation from the hypothesized μ after removing the treatment effect = Σ(X-M)2 • deviation caused by the treatment = variability accounted for (by treatment effect) = (SS1-SS2) • treatment removed from each score: (M-μ ) SS2 {[ X (M )] }2 ( X M )2 ( X M )2 Measuring the percentage of variance explained • indifferent: μ = 10, but M = 13, ∆= M - μ = 3 • How much of the variability is explained by treatment effect? (a) SS1 = Σ(X-μ)2 = 153 :total variability from μ (b) SS2 = Σ(X-M)2 = 72 :adjusted variability from μ (treatment effect removed, i.e. remove the ∆ ) variability accounted for = SS1 - SS2 = 81 r 2 2 2 ( X ) ( X M ) 2 ( X ) SS1 SS2 SS1 SS2 {[ X (M )] }2 ( X M )2 ( X M )2 Percentage of variance accounted for by treatment 2 • r2 vs t2 : t r 2 t df 2 SS1 ( X )2 [( X M ) (M )]2 ( X M )2 n(M )2 SS1 SS 2 ( X M ) 2 n( M ) 2 ( X M ) 2 n( M ) 2 r 2 2 SS1 ( X M ) n( M ) ( X M ) 2 n( M ) 2 2 (M )2 n df ( M ) 2 t 2 ( X M ) n df ( X M ) 2 2 t2 n( M ) 2 2 t df ( X M ) 2 n(M ) 2 Measuring Effect Size for the t Statistic (cont’d.) • By measuring the amount of variability that can be attributed to the treatment, we obtain a measure of the size of the treatment effect. For the t statistic hypothesis test: percentage of variance accounted for = r2 t2 = ───── t2 + df Cohen’s d and r2 • • • • d : not influenced by n d : will be affected by s s↑ d↓ r2 : only slightly affected by n • • • • Interpreting r2 : (proposed by Cohen) r2 = 0.01 small effect r2 = 0.09 medium effect r2 = 0.25 large effect Measuring Effect Size for the t Statistic (cont’d.) • The size of a treatment effect can also be described by computing an estimate of the unknown population mean after treatment. • A confidence interval is a range of values that estimates the unknown population mean by estimating the t value. • A confidence interval is also an interval estimate of the population mean. • A 95% confidence interval means 95% of “confidence” that μ is in that interval. Measuring Effect Size for the t Statistic (cont’d.) • The first step is to select a level of confidence and look up the corresponding t values in the t distribution table. • This value, along with M and sM obtained from the sample data, will be plugged into the estimation formula: μ = M ± tsM M t sM M t sM Example 9.3 (p. 301) • n = 9, df = 8, M = 13, SM = 1, check t table: 80% confidence interval, t = 1.397 13 1.397 *1 80% confidence interval = (11.603, 14.397) 90% confidence interval, t = 1.86 13 1.86 *1 95% confidence interval, t = 2.306 13 2.306 *1 99% confidence interval, t = 3.355 13 3.355 *1 • We assume that the t distribution is exactly what we’ve constructed, so 80% of the probability that true population mean is in this interval. Confidence Interval • confidence level ↑ interval width ↑ • n ↑ sM ↓ interval width ↓ i.e. n ↑ more info more precise estimate M t sM M t sM Report about your test result • p. 302-303: provide info about sample statistics and test statistics and test result/interpretation, eg. M, n, s, t, p, r2 , confidence interval etc. • report an exact p value if you can, otherwise report p < , or p > etc. p. 303 1. n=16, μ=80, M = 86, s=8, α=0.05 sM =8/4=2 a. two-tailed test, significant? df = 15, critical t = 2.131 t = 6/2=3 > 2.131 significant b. Cohen’s d and r2 = ? d = (M- μ)/s=6/8 = 0.75 r2 = 9/(9+15)=0.375 c. 95% CI=? 86 2.131*2 = (81.738, 90.262) 2. n↑ sM ↓ t↑ more likely to reject H0 , but no effect on effect size d s ↑ sM ↑ t↓ less likely to reject H0 , effect size d ↓ Directional / One-tailed test Example 9.4 (p. 304) infant preference M = 13, n = 9, SS = 72 sM = 3/3 = 1 H1: μ>10, α = 0.01 df = 8, critical region: t > 2.896 test statistic t = (13-10)/1 = 3 > 2.896 reject H0 infant spend significantly more time looking at the attractive face • Note: one-tailed test has a larger rejection region more likely to reject H0 than twotailed test with the same α p. 306 normal: μ = 200, n = 9, M = 206, SS = 648, α=0.05 a. H1: μ ≠ 200 critical t = 2.306 s = 648/8 = 81, sM = 9/3 = 3 t = 6/3 = 2 < 2.306 failed to reject H0 b. t(8) = 2, p > 0.05 no significant effect c. H1: μ > 200 critical t = 1.86 t = 6/3 = 2 >1.86 reject H0 d. t(8) = 2, p > 0.05 significantly increases the reaction time