MSAV 662 Exam 2 (Chapters 8, 9, 10, 11, 12)

effect

any outcome or phenomenon of interest in science

effect of interest

typically the dependent variable in a research study, expected to be as informative as possible

hypothesis

a statement or proposed explanation for an observation, a phenomenon, or a scientific problem that can be tested using research methods, often a statement about the value for a parameter in a population, such as the population mean

null hypothesis significance testing (NHST) / hypothesis testing

the method of evaluating statistics in a sample to test hypotheses about parameters in a given population, where we test a hypothesis by determining the likelihood that a sample statistic would be selected if the hypothesis regarding the population parameter were true

Four Steps to Hypothesis Testing

Step 1: State the hypotheses

Step 2: Set the criteria for a decision

Step 3: Compute the test statistic

Step 4: Make a decision

null hypothesis (H₀)

a statement about a population parameter, such as the population mean, that is assumed to be true

alternative hypothesis (H₁)

what we think is wrong about the null hypothesis, directly contradicting a null hypothesis by stating that the actual value of a population parameter is less than, greater than, or not equal to the value stated in the null hypothesis

level of significance

a criterion upon which a decision is made regarding the value stated in a null hypothesis, based on the probability of obtaining a statistic measured in a sample if the value stated in the null hypothesis is true

test statistic

a mathematical formula that identifies how far or how many standard deviations a sample outcome is from the value stated in a null hypothesis, which allows researchers to determine the likelihood of obtaining a sample outcome if the null hypothesis were true

the larger the value of the test statistic

the farther the distance, or number of standard deviations, a sample mean outcome is from the population mean stated in the null hypothesis

if the probability of obtaining a sample mean is less than or equal to 5% when the null hypothesis is true

the decision is to reject the null hypothesis

if the probability of obtaining a sample mean is greater than 5% when the null hypothesis is true

the decision is to fail to reject the null hypothesis

p value

the probability of obtaining a sample outcome, given that the value stated in the null hypothesis is true, which varies between 0 and 1 and can never be negative

decision to reject the null hypothesis

the sample mean is associated with a low probability of occurrence when the null hypothesis is true, and we conclude that the value stated in the null hypothesis is wrong (p < .05) and there is an effect in theb population

decision to fail to reject the null hypothesis

the sample mean is associated with a high probability of occurrence when the null hypothesis is true, so we conclude that there is insufficient evidence to reject the null hypothesis; this does not mean that the null hypothesis is correct, so it is not possible to prove the null hypothesis (p > .05) and there is not an effect in the population

statistical significance

the decision to reject or fail to reject the null hypothesis

null result / null finding

term for the decision to fail to reject the null hypothesis, rarely published in scientific journals for behavioral research due to not being novel enough

Type II error / beta error

term for a “false negative” finding, which is the probability of failing to reject a null hypothesis that is actually false

Type I error

term for a “false positive” finding, which is the probability of rejecting a null hypothesis that is actually true

alpha level (α)

the level of significance or criterion for a hypothesis test that is set, which is the largest probability of committing a Type I error that will be allowed and still decide to reject the null hypothesis, usually set at .05 in behavioral research

onesample z test

a statistical procedure used to test hypotheses concerning a mean value measured in a sample compared to a known value in the population in which the variance in the population is known

nondirectional / two-tailed test

a hypothesis test in which the alternative hypothesis is stated as not equal to (≠) a value stated in the null hypothesis; typically more conservative and makes it more difficult to reject the null hypothesis, hence, the researcher is interested in any alternative to the null hypothesis

critical values

cutoffs which define the boundaries beyond which less than 5% of sample means can be obtained if the null hypothesis is true

rejection regions

the regions beyond the critical values, where if the value of the test statistic falls in, then the decision is to reject the null hypothesis

z statistic

the test statistic for a one-sample z test, which is an inferential statistic that converts any sampling distribution to a standard normal distribution, used to determine the number of standard deviations, or z scores, in a standard normal distribution that a sample mean deviates from the population mean stated in the null hypothesis

obtained value

the value of the z statistic

z statistic formula

the sample mean minus the population mean stated in the null hypothesis, divided by the standard error of the mean

find the p value for the z statistic

find probability (toward the tail) in the unit normal table and multiply by the number of tails for alpha

directional test (one-tailed test)

alternative to the nondirectional test, associated with greater power, which is a hypothesis test in which the alternative hypothesis is stated as greater than (>) or less than (<) a value stated in the null hypothesis

lower-tail critical test (“less than” statement)

level of significance or critical value placed in the lower tail of the sampling distribution

Type III error

a type of error possible with one-tailed tests in which a decision would have been to reject the null hypothesis, but the researcher decides to fail to reject the null hypothesis because the rejection region was located in the wrong tail, referring to the opposite tail from where a difference was observed and would have otherwise been significant

effect size

a statistical measure of the size of an effect in a population, typically describing

1. how far scores shifted in the population, or

2. the percentage of variance that can be explained by a given variable

Cohen’s d

a measure of effect size in terms of the number of standard deviations an effect is shifted above or below a value stated by the null hypothesis, where the larger the absolute value, the larger the effect in the population

(M − μ) / σ

Cohen’s effect size conventions

standard rules for identifying small, medium, and large effects based on typical findings in behavioral research, used to interpret values of d

estimation

a statistical procedure in which a sample statistic is used to estimate the value of an unknown population parameter

point estimate

the use of a sample statistic to estimate a population parameter, such as the value of a sample mean in a single sample; usually associated with low certainty when used by itself

interval estimate

the interval or range of possible values within which a population parameter is likely to be contained; usually associated with greater certainty as it specifies the likelihood that any of the values in the interval contains the unknown population mean

confidence interval (CI)

the interval or range of possible values within which an unknown population parameter is likely to be contained

confidence limits

the upper and lower boundaries of a confidence interval

level of confidence

the probability or likelihood that an interval estimate contains an unknown population parameter; 95% most commonly used

estimation formula for the one-sample z test

M ± z(σM)

steps to estimate the value of a population mean using a point estimate and an interval estimate

1: Compute the sample mean and standard error

2: Choose the level of confidence and find the critical values at that level of confidence.

3: Compute the estimation formula to find the confidence limits

power

the likelihood of detecting an effect, critical because it informs the researcher of the probability that a randomly selected sample will lead to a decision to reject the null hypothesis, if the null hypothesis is false

Factors That Increase Power

• Increasing the alpha level

• Decreasing beta error (β)

• A population with a smaller standard deviation (σ)

• A smaller standard error (σM)

parametric tests

hypothesis tests used to test hypotheses about parameters in a population in which the data in the population are normally distributed and the data are measured on an interval or ratio scale of measurement

skewness

a measure for the symmetry of a distribution and identifies whether the curve of the distribution is pulled or skewed toward the left or right tail relate to a normal distribution

kurtosis

a measure for the “peakedness” of a distribution and identifies whether the data are heavy-tailed (i.e., contain many outliers) or light-tailed (contain very few outliers) relative to a normal distribution

box and whisker plot

a type of graph where

• the box itself represents the upper and lower quartiles or the interquartile range

• a vertical line inside the box marks the median

• “whiskers” typically represent the range of data within a specified distance from the lower and upper quartiles

• any data not included between the whiskers is plotted outside the whiskers and indicates that an outlier is present

nonparametric tests

alternative tests that are used to

(1) test hypotheses that do not make inferences about parameters in a population

(2) test hypotheses about data that can generally have any type of distribution

(3) analyze data on a nominal or ordinal scale of measurement

can be used even when we do not make inferences about parameters in a population, although they can be used to test hypothesized relationships in a population, and generally do not require that the data in the population be normally distributed as the sample size increases; also can be used to analyze data on a nominal or ordinal scale of measurement

estimated standard error (SM)

an alternative to the z statistic, which is an estimate of the standard error or standard distance, in standard deviation units, that sample means deviate from the value of the population mean stated in the null hypothesis

= M = √s² / n = SD / √n

t statistic (t observed / t obtained)

an inferential statistic used to determine the number of standard deviations in a t distribution that a sample mean deviates from the mean value or mean difference stated in the null hypothesis

= (M − μ) / sM , where SM = SD / √n

t distribution (Student’s t)

a normal-like distribution with greater variability in the tails than a normal distribution because the sample variance is substituted for the population variance to estimate the standard error in this distribution; has greater variability in the tails because the sample variance is not always equal to the population variance

degrees of freedom (df)

the number of scores overall, per group, per pairs of scores, or per variable that are free to vary

df for a one-sample t-test

equal to the number of scores overall that are free to vary: n − 1

t table

one-sample t test

a statistical procedure used to test hypotheses concerning a mean value measured in a sample compared to a known value in the population when the variance in the population is unknown

assumptions of a one-sample t test

1. Normality

2. Random sampling

3. Independence

estimated Cohen’s d

a measure of effect size in terms of the number of standard deviations that mean scores shift above or below the population mean stated by the null hypothesis

= (M − μ) / SD

proportion of variance

a measure of effect size in terms of the proportion, or percentage, of variability in a dependent variable that can be explained or accounted for by a treatment

= variability explained / total variability

treatment

in hypothesis treatment, any unique characteristic of a sample, or any unique way that a researcher treats a sample, that can change the value of a dependent variable

Eta-squared (η²)

a measure of the proportion of variance in a dependent variable that can be explained by a treatment, where for the t test, its computation can be expressed in a single formula based on the result:

t² / (t² + df)

although popular, it can overestimate the proportion of variance

Omega-squared (ω²)

a measure of the proportion of variance in a dependent variable that can be explained by a treatment, where for the t test, its computation is expressed in a formula where 1 is subtracted from t² in the numerator to reduce the estimate of effect size:

(t² - 1) / (t² + df)

gives a smaller, more conservative estimate of proportion of variance

estimation formula for the one-sample t test

M ± t(SM)