PSYC 60 Quiz 2 (Ch.7-11) PDF

Ch. 7 Sampling Distributions ​ Sampling distribution: a probability distribution of all statistics (e.g. all sample means) of a given sample size from a population. ○​ The sampling distribution of the mean is the most common sampling distribution ​ Mean...

Ch. 7 Sampling Distributions ​ Sampling distribution: a probability distribution of all statistics (e.g. all sample means) of a given sample size from a population. ○​ The sampling distribution of the mean is the most common sampling distribution ​ Mean of a sampling distribution of the mean: μM ​ The variance of the sampling distribution of the mean is denoted: σ²M ​ Standard error: the standard deviation of a sampling distribution ​ Measure of sampling error ○​ Sampling error: the difference between a statistic and the parameter ​ It’s the variability of a statistic from sample to sample due to chance; not mistakes in sampling. ​ The bigger the standard error the bigger the sampling error.** ○​ The standard error changes with sample size and the population standard deviation. ​ A smaller population standard deviation equals a smaller standard error. ​ The less that population scores deviate from the mean, the less that sample scores can deviate from the population mean. ​ Increasing the sample size will decrease standard error (decrease sampling error) because of the law of large numbers. ○​ Law of large numbers: the law that increasing sample size decreases standard error. ​ The larger the sample size, the more likely the sample’s statistics will resemble the corresponding parameters. ​ Hence, larger samples are associated with more accurate estimates of parameters. ​ Central limit theorem: a mathematical statement regarding the nature of sampling distributions ○​ The theorem is that given a population with a mean μ and a variance σ², the sampling distribution of the mean will: ​ Have a mean equal to μ ​ Have a variance equal to σ²/N (N = the sample size, NOT the number of samples)  ​ Approach the normal distribution as N increases, regardless of the shape of the population. ○​ Ex: ○​ The central Limit theorem is important because it allows for inferential statistics, ex, allows scientists to draw conclusions about populations from samples. ** Ch. 8 Hypothesis Testing Null Hypothesis Significance Testing ​ Hypothesis: a statement, e.g., the value of a parameter, or a tentative explanation of a phenomenon ○​ Example statement: the population mean final grade for undergraduate statistics is 80% ○​ Example tentative explanation: statistics is challenging to learn because the concepts are often not intuitive. ​ Null hypothesis (H0): a statement about a parameter,e.g., a population mean, that is ASSUMED to be true. ○​ It's usually a hypothesis of no difference or no relationship between variables (innocent until proven guilty) ○​ E.g., a null hypothesis could be that there is NO relationship between studying and test performance. ○​ The alternative to a null hypothesis is the alternative hypothesis. ​ Alternative/research hypothesis (H1): a statement that directly contradicts a null hypothesis.  ○​ It’s the hypothesis that usually the researcher BELIEVES TO BE TRUE ○​ E.g., an alternative hypothesis could be that there IS a relationship between studying and test performance. ○​ H0 and H1 must encompass ALL possibilities for a population, e.g., studying either is or is not related to test performance. ○​ The researcher’s aim (in almost all cases) is to reject the null hypothesis ​ Null hypothesis (H0) Alternative/research hypothesis (H1) There is NOT a relationship between There IS a relationship between studying and test performance. studying and test performance. ​ Null hypothesis significance testing (NHST): the evaluation of statistics to estimate parameters. ○​ NHST calculates the probability of obtaining a statistic if the hypothesis regarding the parameter is true. ○​ STEPS: ​ (1) State the hypothesis, i.e., state the null hypothesis (H0) and research hypothesis (H1) ​ (2) Set the criteria for rejecting/not rejecting the null hypothesis ​ (3) Compute the test statistic, i.e., calculate the probability of obtaining a statistic if H0 is true. ​ (4) Make a decision; ex: reject or do not reject the null hypothesis. One- and Two-Tailed Tests ​ Hypotheses are often evaluated with one- or two- tailed tests. One-tailed (directional) test Two-tailed (nondirectional test) A hypothesis test in which the value in A hypothesis test in which the value in H1 is states as > OR < the value in H0 H1 is stated as ≠ the value in H0 Ex: the population mean final grade Ex: the population mean final grade for undergraduate statistics is > 80% for undergraduate statistics ≠ 80% Ex: the population mean final grade for undergraduate statistics is < 80% ​ Alpha (α): the significance level for a hypothesis test ○​ It’s usually set at 0.05 (α = 0.05) in behavioral research  ○​ 5% of the distribution will serve as the rejection region. Calculating a Test Statistic ​ Test statistic: a formula to determine the probability of obtaining a statistic if the null hypothesis is true. ○​ If a test statistic is GREATER than a critical value, then REJECT the null hypothesis. ​ Critical value: a cutoff value ○​ Ex: the critical values on slide 18 = ±1.96. Thus, a test statistic that is < -1.96 or > +1.96 would indicate to REJECT H0 ​ One-sample z test: an inferential statistic that uses z scores to determine if a sample mean is significantly different from a population mean ○​ ○​ ○​ P value ​ P value: the probability of obtaining a test statistic that is at least as extreme as the observed result if H0 is true.  ○​ P values determine whether H0 should be rejected. ○​ If p < α, REJECT H0 ○​ If p > α, DO NOT REJECT H0 Type I and Type II Errors ​ Type I error: the error of rejecting H0 when H0 is true (false positive) ○​ The probability of making a Type I error = α ​ Type II error: the error of NOT REJECTING H0 when H0 is false ○​ The probability of making a Type II error = β ​ Power: the probability of correctly rejecting a false H0 (THIS IS GOOD) ○​ The probability = 1- β ​ Increasing the Type I error rate decreases the type II error rate, and vice versa, hence we don’t set either to zero. ​ Ch. 9 One-sample T test Definition and Examples ​ T test: an inferential statistic for determining if there is a significance difference between two means. ○​ T tests include one sample t tests, dependent-samples t tests, and independent-samples t tests ​ One-sample t test: an inferential statistic for determining if there is a significant difference between a sample mean and population mean. ​ Example questions include for ONE-SAMPLE T TEST: ○​ Is there a difference in graduate record examination (GRE) scores between a sample of Californians and the population of Americans? ○​ Is there a difference in the personality trait agreeableness between a sample of 20-year-olds and the general population. ○​ Is there a difference in working memory capacity between a sample of STEM students and the general population? ​ ○​ The formula for the standard error of the mean is the denominator of the one-sample z test: (σ/√N) ○​ The denominator of the one-sample t test (SD/√N) provides an estimate of the standard error of the mean. ​ Z tests use the standard normal distribution; t tests use the t distribution (df increase with sample size) ○​ Assumptions ​ Two assumptions must be met for one-sample t tests to be valid: ○​ (1) The DV is measured with a ratio or interval scale. ○​ (2) The sample is drawn from a normally distributed population. ​ Normality is met: ○​ Skewness coefficient is between ±1 (SYMMETRY) ​ Positive coefficient (i.e.,> 0) for skew indicates positive skew. ​ ​ Negative coefficient (i.e., < 0) for skew indicates negative skew. ​ ​ Coefficient of 0 for skew indicates a symmetrical distribution. ○​ Kurtosis coefficient is between ±2 (POINTEDNESS) ​ Kurtosis of the standard normal distribution is 3 (mathematically), but set to 0 in SPSS ​ Positive coefficients for kurtosis indicate a more pointed distribution, i.e., leptokurtic ​ Negative coefficients for kurtosis indicate a less pointed distribution, i.e., platykurtic. ​ Standard normal distribution is mesokurtic  ​ Calculating t ​ ​  ​ ​ ​ Effect Size ​ Effect size: a measure of the magnitude of a relationship between variables ○​ Many effects are statistically significant with large enough samples but not necessarily meaningful or important. ○​ Effect size informs if a significant result is meaningful or important. ​ Cohen’s d: a measure of effect size in terms of the number of standard deviations that mean differ from each other. (estimated) ○​ The most common measure of effect size for t tests ○​ It’s estimated because statistics are used instead of parameters ○​ Example:  ​ ​ Negatives could happen when sample mean is smaller than population mean! ○​ Interpreting Cohen’s D value: ​ ○​ In other words using the previous example except with N = 25… ​ Confidence Interval ​ Confidence interval (CI): a range of values that likely contains an unknown parameter. ○​ Calculating a confidence interval provides a way to: ​ Estimate μ (in our case, μ for STEM students) ​ Infer statistical significance ​ Estimate effect size ○​ Example:  ​ ​ Interpretation: At a 95% confidence interval, the estimated population mean of STEM students’ working memory capacity is between 4.09 and 4.91 Factors that Influence Power ​ Power: the probability of correctly rejecting a false H0 (THIS IS GOOD) ○​ The probability = 1- β Factors that influence power Meaning Effect size Effect increase, power increase -​ As the difference between M and μ increases, power increases Sample size N increase, decrease sampling power and increase power -​ Imagine a sample size increasing to the point it includes the entire population. Sampling error would decrease 0. M would equal μ Alpha (α) Increasing α, will increase power Beta (β) Increasing β, will decrease power -​ 1-β = power; therefore, increasing beta decreases power. Standard Deviation (SD) As SD increases, power decreases -​ Populations with more dispersion of scores produce samples with more dispersion. -​ More dispersion equals more sampling error, and less certainty that samples are representative of populations. Ch. 10 Independent-Samples t Test Definition and Examples ​ Independent-samples t test: an inferential statistic for determining if there is a significant difference between two independent/unrelated samples. ○​ Independent-samples t tests are a type of between-subject designs. ​ Between-subjects design: a research method in which participants are assigned to only one of multiple conditions, and then scores are obtained and compared across conditions. ​ Examples question for INDEPENDENT-SAMPLES T TESTS: ○​ Is there a difference in graduate record examination (GRE) scores between a sample of Californians and a sample of Delawareans? ○​ Is there a difference in the personality trait agreeableness between a sample of 20-year olds and a sample of 60-years-olds? ○​ Is there a difference in working memory capacity between a sample of STEM students and a sample of art students? Assumptions ​ Three assumption must be met for independent-samples t tests to be valid: ○​ (1) The DV is measured with a ratio or interval scale. ○​ (2) The samples are drawn from normally distributed populations ○​ (3) The samples are drawn from populations with equal (homogenous) variances. Calculating T ​ ​ ​ ​ ​ ​ **The denominator is the estimated standard error for the difference between means.** ​ ​ ​ Effect Size ​ ​ Confidence Interval ​ For Independent samples, the CI provides a way to: ○​ Estimate the difference between μ1 and μ2 ​  ​ ○​ Infer statistical significance (DON'T HAVE TO DO ON TEST) ○​ Estimate effect size (DON'T HAVE TO DO ON TEST) Ch. 11 Dependent-Samples t Test Definition and Examples ​ Dependent-samples T test: an inferential statistic for determining if there is a significant difference between two dependent/related samples. ○​ Dependent-samples t tests are a type of within-subjects design. ​ Within-subjects design: a research method in which participants are assigned to multiple conditions, and then scores are obtained and compared across conditions. ​ Example questions for DEPENDENT-SAMPLES T TESTS: ○​ Is there a difference between in a sample of individuals’ graduate record examination (GRE) scores from before to after a GRE prep course? ○​ Is there a difference in a sample of 20-year-olds’ agreeableness scores from two different personality measures? ○​ Is there a difference in a sample of STEM students’ working memory capacity from before to after caffeine intake? Assumptions ​ Two assumptions must be met for dependent-sample t tests to be valid: ○​ (1) the DV is measured with a ratio or interval scale ○​ (2) The samples are drawn from normally distributed populations Calculating T ​ ​ ​ ​ ○​ Subscript “d” is the difference of the “before” and “after” scores ○​ The denominator is the estimated standard error for the mean difference scores. ​ ​ ​ Effect Size ​ ​ Confidence Interval ​ For dependent samples, the CI provides a way to: ○​ Estimate μD ​  ​ ○​ Infer statistical significance (DON'T HAVE TO DO ON TEST) ○​ Estimate effect size (DON'T HAVE TO DO ON TEST) Advantages of Within-Subjects Design ​ The advantages of within-subjects (w/n-Ss) designs vs. between-subjects (b/n-Ss) designs are that the former increase: ○​ (1) power, by decreasing standard error by: ​ (a) eliminating person-to-person variability ​ Ex: ​ ​ ALTERNATIVELY SUPPOSE: ​ ​ (b) increasing control of extraneous variables ​ Ex:  ​ ​ ALTERNATIVELY SUPPOSE: ​ ○​ (2) practicality ​ Recruiting participants require time, energy, and often money ​ For the same amount of power as a b/n-Ss design, w/n-Ss design requires fewer participants, and thus costs less time, energy, and money.