Lecture13-February+25th-Hypo+testing2 (1)

# Lecture13-February+25th-Hypo+testing2 (1) -  ...

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Unformatted text preview:   Next week    Last tutorial due    Exercise 4 due    Z‐test = Testing the Null hypothesis on single  sample    Statistics estimates population parameters    Statistics will always diﬀer from parameters due  to “sampling error”    How can we test whether our “treated” sample  diﬀers “signiﬁcantly” from population?    Population: PSC41 students     DV= test scores    μ = 38, σ = 4    Sample: 25 students from PSC41    IV= practice questions; DV= test scores    M= 41 (or 39)    Is this group signiﬁcantly diﬀerent from population? Does  our manipulation have an eﬀect?    Rationale: what would be the probability of  getting the same mean if we selected all of  possible samples (of same size) from the  population?     What are the odds that our result is due to  chance?    Test of null hypothesis:   ▪  H0: μpractice = 38    We need to know the “sampling distribution”    Distribution of “mean score” (on our DV) of all the  possible samples (of 25 students) that could be  drawn from the population    Sampling distribution has  ▪  Mean = mean of population μ  ▪  SD =  σ/√n (standard error: variation between samples)    Sampling distribution quickly approaches  normality (with increasing sample size)    As sampling distribution is normal, we can  standardize scores to get the probability  associated with their occurrence (with the  occurrence of each sample)    Using z‐scores we can ﬁnd the probability  associated with any speciﬁc sample    The location of our sample mean z=    68% 95% 99% (sample mean‐μ)    σ/√n When will we  consider our  sample to be  signiﬁcantly  diﬀerent from a  random sample?     Level of signiﬁcance: α    Determines the sample means that could be  obtained with the lowest probability    68% 95% 99% If H0 is true, most  sample means  should be close  to distribution  mean    α: conventionally set at 5% and 1%  The distribution of sample means if the null hypothesis is true (all the possible outcomes) Reject H0 Extreme, low-probability values if H0 is true Sample means close to H0: High probability values if H0 is true µ from H0 Reject H0 Extreme, low-probability values if H0 is true   One‐tailed vs. Two tailed test    For a directional alternative hypothesis we’ll  conduct a one tailed test  Critical Region ▪  H0: μ < 38  ▪  H1: μ > 38    For a non‐directional alternative hypothesis we’ll  conduct a two‐tailed test  ▪  H0: μ = 38  ▪  H1: μ ≠ 38  Critical Region Critical Region   Distribution of z‐scores : + 1.96 correspond to  cutoﬀ scores for critical region (2‐tailed)      If z‐score for our sample  falls between ‐1.96 and  +1.96, we fail to reject H0  If z‐score for our sample  Z=-1.96 falls below ‐1.96 or +1.96,  Reject H0 we reject H0  Z= (41-38) 4/√25 = 3.75 Middle 95%: high-prob values if H0 is true µ from H0 Z=1.96 Reject H0 Critical Region: Extreme 5%   Statistical Signiﬁcance test  1.  Calculate critical values set by α  2.  Calculate probability of   getting your sample mean:  p‐value  ▪  If p‐value < α then   reject null hypothesis  ▪  If p‐value > α then   fail to reject null hypothesis  P-value for M = 39 P-value: Likelihood of getting the sample mean that we got if H0 is true P-value for M = 41   P‐value: probability of getting a test statistic  (i.e., mean) AT LEAST as extreme as the one  observed, assuming H0 correct    Psc41: μ=38; Practice sample: M=41    Z=3.75,  p<.001  ▪  Probability of practice sample performing the way they  did, assuming there’s no diﬀerence b/w them and the  population they were drawn from, is less than 1%  ▪  The lower the p‐value, the more "signiﬁcant" the result, in  the sense of statistical signiﬁcance (based on α)    REMEMBER:    p > α means H0 cannot be rejected AT THAT  SIGNIFICANCE LEVEL. Does not mean H0 is true    Small p‐values do not IMPLY that H1 is correct    Signiﬁcance level is NOT determine by p‐value. It is  decided upon BEFORE viewing data, and is  COMPARED to p‐value     α= .05 is a convention!    P‐value is not indication of importance or magnitude  of an eﬀect    “…First, a main eﬀect of expected memorability  was observed, speciﬁcally  higher false‐alarm  rates for low compared to high expected‐ memorability events (Low: M = .20, SD = .13;  High: M = .17, SD = .12), F (1, 96) = 10.22, p < .01,  ηp2 = .10...”  Descriptive Stats Inferential Stats p value if H0 is true   Errors are possible in the statistical decision  process    Type I error  ▪  Rejecting null hypothesis that is actually true  ▪  Always equals to the alpha level    Type II error  ▪  Failing to reject a null hypothesis that is really false  True State of Affairs Ho True Reject Ho Decision Do not reject Ho Ho False Type I error Correct decision (= α ) Correct decision (confidence level=1- α) (power=1- β) Type II error (=β )   REMEMBER:    p > α means H0 cannot be rejected AT THAT  SIGNIFICANCE LEVEL. Does not mean H0 is true    Failure to ﬁnd signiﬁcant eﬀect, simply means you  didn’t ﬁnd the eﬀect BASED ON YOUR STUDY    It is possible you didn’t because your statistical  POWER was low    Power    Probability that the test will correctly reject a false  null hypothesis  ▪  Sensitivity of stat procedure to ﬁnd the hypothesized  diﬀerences    Increase power by larger sample size    Increase power by rigorous research design  ▪  Reliability and validity!    Signiﬁcance test: what are the odds that the  detected diﬀerence is due to chance?    Eﬀect size: is the diﬀerence meaningful?    Measure the absolute magnitude of a treatment  independent of the sample size    How substantial is the eﬀect?    “…First, a main eﬀect of expected memorability  was observed, speciﬁcally  higher false‐alarm  rates for low compared to high expected‐ memorability events (Low: M = .20, SD = .13;  High: M = .17, SD = .12), F (1, 96) = 10.22, p < .01,  ηp2 = .10...”  Effect size Inferential Stats p value if H0 is true   The t‐test    For single sample    For independent groups    For correlated groups    Recap    The sample mean (noted as M or x)  is expected to  be equal to the population mean (μ)    The standard error measures how well a sample  mean approximate the population mean    Compare M to μ by computing a z‐score test  statistic  Problem: usually unknown (M - µ ) z= σ/√n = obtained difference between data and hypothesis Standard distance between data and hypothesis   An alternative to Z test  (M - µ ) z= σ/√n (M - µ ) t= s/√n The t statistic   Testing hypotheses about an unknown population  mean μ when the value of σ is unknown    T test    Data are interval or ratio    Population distribution is symmetrical    mean μ is hypothesized but value of σ is unknown    We estimate population sd using the sample sd    Means are not distributed normally    Use Student T distribution    Student’s t distribution :    Not one distribution, but diﬀerent, depending on  sample size    Diﬀerent probabilities associated with diﬀerent  samples     Student’s t distribution :    Diﬀerent distributions for diﬀerent sample sizes    df= degrees of freedom  ▪  Sample size (n)‐1  ▪  The # of scores   in the sample that   are independent   and free to vary    Comparing a sample mean to population mean  when is σ unknown    The signiﬁcance test  1.  Calculate critical values set by α: t statistic  ▪  If  ‐tcritical value<tsample mean< tcritical value then you fail to reject H0  ▪  If  tsample mean<‐tcritical value  or tsample mean> tcritical value then reject H0  2.  Calculate probability of getting your sample  mean: p‐value  ▪  If p‐value < α then reject null hypothesis  ▪  If p‐value > α then you fail to reject null hypothesis    This value is used just like a z‐statistic     if the value of t exceeds critical valued, tα , then an  eﬀect is detected (i.e., the null hypothesis is rejected)  31    For a directional alternative hypothesis we’ll  conduct a one tailed test  Critical Region   For a non‐directional alternative hypothesis  we’ll conduct a two‐tailed test  Critical Region Critical Region   The t‐test    For single sample    For independent groups    For correlated groups    Are the two groups diﬀerent?    Are they so similar they could come from  same population?  M − µ0 t= sM Difference b/w group means M1 − M 2 t= sM 1− M 2 Variability b/w groups €   Standard error of diﬀerence between means  s1 s2 s s sM 1 − M 2 = + n1 n 2 2 1 2 2   Are means of two groups signiﬁcantly  diﬀerent from each other?  µ1 = µ2 : µ1 ≠ µ2 H0 : H1 or H1 : µ1 > µ2 or H1 : µ1 < µ2   To ﬁnd level of signiﬁcance we also need to  know the DF    Df for independent sample is N1 +N2 ‐2    With t and df we check whether we can reject  the null hypothesis  µ1 = µ2 : µ1 ≠ µ2 H0 : Reject H1 or H1 : µ1 > µ2 or Ha : µ1 < µ2 As the t statistic & df increase, p-value decreases => more likely to reject null hypothesis   IMPORTANT:    Comparison it at the GROUP level (not the  individual)    The mean are diﬀerent. Does not imply every  member of one group is diﬀerent from every  member of the other group    Correlated/Paired t‐test: Comparing  mean diﬀerence of correlated scores    When do we use it?  1.  In a Matched groups  design   (aka paired comparison study)  2.  In a Within‐participants design:   2 repeated measures or pretest‐posttest    Are means of two groups signiﬁcantly  diﬀerent from each other?  µ1 - µ2 = 0 : µ1 - µ2 > 0 (one‐tailed)  H0 : H1   Sampling distribution would not be of means,  but of diﬀerences b/w means    To calculate t we ﬁrst need to compute  DIFFERENCE SCORES (the diﬀerence  between score in one condition vs. the other)  M − µ0 t= sM MD − 0 t= sD Standard error of difference score €   Standard error of diﬀerence scores  ∑ (D − M sD = N −1 n D ) 2 ...
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