1 PSYC 2021 - Exam 3 Notes Chapter 7: Introduction to ...
1
PSYC 2021 - Exam 3 Notes
Chapter 7: Introduction to Inferential Statistics Standard error of the mean: σμ = σ / √N. The smaller the σμ, the more normal the distribution. Best estimate of σ2 = σ2 + σμ2 = (SS / N) + (SS / N [N – 1]) = SS / (N – 1) Z scores can also be calculated using: z = (Xbar – μ) / σμ2 H0 makes an explicit statement while the H1 makes a directional claim. H1 ≠ H0 z = (Xbar – μ as stated by H0) / σμ2 where H1 is the alpha (the region of rejection) Type I error: H0 is rejected, but H0 is true.
Type II error: H0 is retained, but H0 is false.
Type I is justified, but rejected. Type II is temporarily passive, but wrongfully accepted. Alpha is the risk of making a Type I error. Decreasing Type I risk increases Type II risk. Chapter 8: Testing Hypotheses using a Single Sample Standard error of the mean = √(SS / N[N – 1]) = schevron / √N = s / √(N – 1) Degrees of freedom (df): in the single sample mean, df = N – 1if there is only one sample mean involved in the calculation of the standard error of the mean. Using df is irrelevant to the z-test.
If σ is known, z-test is used.
Choice of alpha: conventional α = .05
Probability of a Type I error: < .01
95% Confidence Interval (C95): μ = Xbar ± tα = .05 x (standard error of mean), where Xbar is the center of the confidence interval and tα defines the width of the interval. Chapter 9: Variables, Within-Subjects Designs, and the T-Test for Correlated Samples Disturbance (noise / random) Variables: the clear relation that ideally exists between IV and DV where other variables tend to muddle this relation. Confounding variables: a systematic correlation between the IV and DV that affect the dependent variable correlated to the IV.
2
Carrying out the t-test for correlated samples: Z = (Xbar - µ) / σxbar
t = (Xbar - µ) / schevronxbar
schevronDbar = √(SSD / N[N - 1])
t = (Dbar - µDbar) / schevronDbar
SSD = ΣD2 – ([ΣD]2 / N)
Steps: (1) State H0 and H1 and determine if the test is one-tailed or two-tailed. (2) Select the alpha level, calculate df, and look up tcritical. (3) Calculate D, D2, ΣD, ΣD2. (4) Calculate Dbar, SSD, schevronDbar. (5) Calculate tobtained. (6) State a conclusion. Chapter 10: t-test for Independent Samples The standard error of the difference between means: σxbar1 – xbar2 = √(σ2 [1/n1 + 1/n2]) The problem is that knowing the true value of σ is a luxury one only tends to encounter in the one sample case. (Schevron12 + Schevron22) / 2 is used to come closer to the true σ2. σxbar1 – xbar2 = √[(Schevron12 + Schevron22) / 2 x (1/n1 + 1/n2)] = √(pooled variance estimate)(1/n1 + 1/n2) Pooled variance estimate is a remedy for the sample sizes by weighing each variance by the df on which it is based instead of simple averaging between two independent sample sizes.
Pooled variance estimate: (SS1 + SS2) / (n1 + n2 – 2)
The nature of the effect is determined through visual inspection of the sample means. The size of the effect is calculated by a measure of association called eta2, which is similar to coefficient of determination r2. Eta is a measure of correlation between and the independent and the dependent variable. Eta2 = t2 / (t2 + df) The t-test for independent groups must be measured on an interval or ratio scale. Chapter 11: The Power of Statistical Tests and the Problem of Hypothesis Testing Leavitt believes that the rejection of the null hypothesis as the real intent in psychological experiments would no longer require any need to gather data. The real problem however, is that
3
the effect of the independent variable that the researcher is claiming to demonstrate is confounded with a powerful factor that has nothing to do with the research question: the size N. Tobtained = (xbar – µ) / sxbar = √N(xbar – µ) / σ
sxbar = σ / √N
The obtained t-value can be increased if the discrepancy between xbar and µis larger while decreasing the sxbar by increasing N. Any difference between groups no matter how minute and trivial, will turn out to be statistically significant if N is sufficiently large. The power of a statistical test is defined as the test’s ability to reject H0 when it is false. We reject H0 when the obtained t-value is larger (>) than the critical t-value.
The factors that affect the power of a statistical test are directionality (one or two tailed), alpha (higher = stronger), degrees of freedom (greater df providing more power than low df), type of test (z-test is powerful than t-test if σ is present), the size of σ, (smaller the value, the fewer disturbance variables), the type of research design within a subject design (homogeneity vs. heterogeneity), and the sample size N (larger N = more power).
Observational research using quasi-experimental design benefits from low costs and selfassignment of subjects to various groups.
The presence of an effect is a less radical approach to hypothesis testing that avoids the confounding effect of a huge N because it contemplates the results. It helps calculate the size of the effect (eta2).
Chapter 15: The Chi Square Test Parametric tests are run very quickly and efficiently, but it can falter if basic conditions are not met. It requires data of the DV be measured on an interval or ratio scale and distributed normally. Non-parametric tests run more slowly than parametric tests, but they are efficient regardless of conditions. It is distribution-free and one subtype of it can run on an ordinal scale.
4
Chi Square (x2)
The other non-parametric test subtype that runs on a nominal scale is called the Chi Square (x2). The frequency of its qualitative values is where the chi square operates.
Historically, Chi square tests were first published by Karl Pearson in 1900. The two different versions of the Chi square test goodness of fit and homogeneity.
Goodness of fit: compares frequency distribution of empirical data to theoretical distribution. x2 = ∑(O – E)2 / E
where expectation (E) and observed frequencies (O) represent the squared
deviations (X – Xbar)2 = (O – E)2 and E = N / k and df = k – 1
(k is the number of total
outcomes and N is the number of trials)
Chi square is highly skewed at low degrees of freedom and is based on positive squares.
Requirements of the Chi square test 1. Only calculated on the frequencies of values; % and proportions must be converted 2. Values must be variable exhaustive (comparing variable p and variable q) 3. Observations must be independent of each other. Only one type of observation observed. 4. Number of expected frequencies never goes below 5, especially if the df is low. It is used to avoid inflation of the obtained x2 value. E can never be zero (infinity will result if zero). Grouping similar categories together can avoid small E values. The goodness of fit test works best if the theoretical distribution is rectangular. Others can work. Clear-cause effect relationships can only be inferred on random assignment to groups. Test of homogeneity: based on two variables measured on the same individual, all observations are independent of each other. It is a test of correlation between the two variables.
Expected frequencies (fe) = (∑row)(∑column) / N = p(cell) x N = (∑row)( ∑column) / N
p(row) = ∑row / N
p(column) = ∑column / N
5
The probability of being in a particular cell p(cell) is a product of the probability of being in the row p(row) and the probability of being in the column p(column).
df = (number of rows – 1)(number of columns – 1)
p(cell) = (∑row / N)( ∑column / N)
The information that an association exists has to be complemented by the information on the nature of the association and the strength of the association.
The strength of the association can be calculated using Cramer’s V = √([x2] / [N{L – 1}]) where L is the number of levels of the variable with the fewest levels.
V must be between -1 and +1 because it is a correlation.
The chi square (x2) is the only test available for nominal scale measurements where variables take on more than two values. It can also be measured on other scales, but it must be simplified.
The t-test for two independent variables can be used to yield the same results. However, the t-test cannot tolerate any test that has more than two independent variables.
PSYC 2021 - Exam 3 Notes
Chapter 7: Introduction to Inferential Statistics Standard error of the mean: σμ = σ / √N. The smaller the σμ, the more normal the distribution. Best estimate of σ2 = σ2 + σμ2 = (SS / N) + (SS / N [N – 1]) = SS / (N – 1) Z scores can also be calculated using: z = (Xbar – μ) / σμ2 H0 makes an explicit statement while the H1 makes a directional claim. H1 ≠ H0 z = (Xbar – μ as stated by H0) / σμ2 where H1 is the alpha (the region of rejection) Type I error: H0 is rejected, but H0 is true.
Type II error: H0 is retained, but H0 is false.
Type I is justified, but rejected. Type II is temporarily passive, but wrongfully accepted. Alpha is the risk of making a Type I error. Decreasing Type I risk increases Type II risk. Chapter 8: Testing Hypotheses using a Single Sample Standard error of the mean = √(SS / N[N – 1]) = schevron / √N = s / √(N – 1) Degrees of freedom (df): in the single sample mean, df = N – 1if there is only one sample mean involved in the calculation of the standard error of the mean. Using df is irrelevant to the z-test.
If σ is known, z-test is used.
Choice of alpha: conventional α = .05
Probability of a Type I error: < .01
95% Confidence Interval (C95): μ = Xbar ± tα = .05 x (standard error of mean), where Xbar is the center of the confidence interval and tα defines the width of the interval. Chapter 9: Variables, Within-Subjects Designs, and the T-Test for Correlated Samples Disturbance (noise / random) Variables: the clear relation that ideally exists between IV and DV where other variables tend to muddle this relation. Confounding variables: a systematic correlation between the IV and DV that affect the dependent variable correlated to the IV.
2
Carrying out the t-test for correlated samples: Z = (Xbar - µ) / σxbar
t = (Xbar - µ) / schevronxbar
schevronDbar = √(SSD / N[N - 1])
t = (Dbar - µDbar) / schevronDbar
SSD = ΣD2 – ([ΣD]2 / N)
Steps: (1) State H0 and H1 and determine if the test is one-tailed or two-tailed. (2) Select the alpha level, calculate df, and look up tcritical. (3) Calculate D, D2, ΣD, ΣD2. (4) Calculate Dbar, SSD, schevronDbar. (5) Calculate tobtained. (6) State a conclusion. Chapter 10: t-test for Independent Samples The standard error of the difference between means: σxbar1 – xbar2 = √(σ2 [1/n1 + 1/n2]) The problem is that knowing the true value of σ is a luxury one only tends to encounter in the one sample case. (Schevron12 + Schevron22) / 2 is used to come closer to the true σ2. σxbar1 – xbar2 = √[(Schevron12 + Schevron22) / 2 x (1/n1 + 1/n2)] = √(pooled variance estimate)(1/n1 + 1/n2) Pooled variance estimate is a remedy for the sample sizes by weighing each variance by the df on which it is based instead of simple averaging between two independent sample sizes.
Pooled variance estimate: (SS1 + SS2) / (n1 + n2 – 2)
The nature of the effect is determined through visual inspection of the sample means. The size of the effect is calculated by a measure of association called eta2, which is similar to coefficient of determination r2. Eta is a measure of correlation between and the independent and the dependent variable. Eta2 = t2 / (t2 + df) The t-test for independent groups must be measured on an interval or ratio scale. Chapter 11: The Power of Statistical Tests and the Problem of Hypothesis Testing Leavitt believes that the rejection of the null hypothesis as the real intent in psychological experiments would no longer require any need to gather data. The real problem however, is that
3
the effect of the independent variable that the researcher is claiming to demonstrate is confounded with a powerful factor that has nothing to do with the research question: the size N. Tobtained = (xbar – µ) / sxbar = √N(xbar – µ) / σ
sxbar = σ / √N
The obtained t-value can be increased if the discrepancy between xbar and µis larger while decreasing the sxbar by increasing N. Any difference between groups no matter how minute and trivial, will turn out to be statistically significant if N is sufficiently large. The power of a statistical test is defined as the test’s ability to reject H0 when it is false. We reject H0 when the obtained t-value is larger (>) than the critical t-value.
The factors that affect the power of a statistical test are directionality (one or two tailed), alpha (higher = stronger), degrees of freedom (greater df providing more power than low df), type of test (z-test is powerful than t-test if σ is present), the size of σ, (smaller the value, the fewer disturbance variables), the type of research design within a subject design (homogeneity vs. heterogeneity), and the sample size N (larger N = more power).
Observational research using quasi-experimental design benefits from low costs and selfassignment of subjects to various groups.
The presence of an effect is a less radical approach to hypothesis testing that avoids the confounding effect of a huge N because it contemplates the results. It helps calculate the size of the effect (eta2).
Chapter 15: The Chi Square Test Parametric tests are run very quickly and efficiently, but it can falter if basic conditions are not met. It requires data of the DV be measured on an interval or ratio scale and distributed normally. Non-parametric tests run more slowly than parametric tests, but they are efficient regardless of conditions. It is distribution-free and one subtype of it can run on an ordinal scale.
4
Chi Square (x2)
The other non-parametric test subtype that runs on a nominal scale is called the Chi Square (x2). The frequency of its qualitative values is where the chi square operates.
Historically, Chi square tests were first published by Karl Pearson in 1900. The two different versions of the Chi square test goodness of fit and homogeneity.
Goodness of fit: compares frequency distribution of empirical data to theoretical distribution. x2 = ∑(O – E)2 / E
where expectation (E) and observed frequencies (O) represent the squared
deviations (X – Xbar)2 = (O – E)2 and E = N / k and df = k – 1
(k is the number of total
outcomes and N is the number of trials)
Chi square is highly skewed at low degrees of freedom and is based on positive squares.
Requirements of the Chi square test 1. Only calculated on the frequencies of values; % and proportions must be converted 2. Values must be variable exhaustive (comparing variable p and variable q) 3. Observations must be independent of each other. Only one type of observation observed. 4. Number of expected frequencies never goes below 5, especially if the df is low. It is used to avoid inflation of the obtained x2 value. E can never be zero (infinity will result if zero). Grouping similar categories together can avoid small E values. The goodness of fit test works best if the theoretical distribution is rectangular. Others can work. Clear-cause effect relationships can only be inferred on random assignment to groups. Test of homogeneity: based on two variables measured on the same individual, all observations are independent of each other. It is a test of correlation between the two variables.
Expected frequencies (fe) = (∑row)(∑column) / N = p(cell) x N = (∑row)( ∑column) / N
p(row) = ∑row / N
p(column) = ∑column / N
5
The probability of being in a particular cell p(cell) is a product of the probability of being in the row p(row) and the probability of being in the column p(column).
df = (number of rows – 1)(number of columns – 1)
p(cell) = (∑row / N)( ∑column / N)
The information that an association exists has to be complemented by the information on the nature of the association and the strength of the association.
The strength of the association can be calculated using Cramer’s V = √([x2] / [N{L – 1}]) where L is the number of levels of the variable with the fewest levels.
V must be between -1 and +1 because it is a correlation.
The chi square (x2) is the only test available for nominal scale measurements where variables take on more than two values. It can also be measured on other scales, but it must be simplified.
The t-test for two independent variables can be used to yield the same results. However, the t-test cannot tolerate any test that has more than two independent variables.
