Study Guide #2
Nominal Scale
A classification system that places people, objects, or other entities into mutually exclusive categories. Does not provide any quantitive information. Uses classification variables (ex: nationality/subject sex)
Research Hypothesis
Dependent variable appears on the left and the independent variable appears on the right A description of the differences or relationships that you predict you will find at the conclusion of your study (should be based on previous theory and research)
Old School Approach to displaying the results of a significance test
"My results were statistically significant, z = +2.31, p < .05" or if you set alpha at .01..z = +3.45, p < .01 or if they are nonsignifcant "My results were statistically nonsignificant, z = +1.25, p > .05
When computing the 95% confidence interval following a single-sample z test, what is the critical value of the z statistic that should be used in the computation of the confidence interval?
+/- 1.96
Type II Error
-Failing to reject the null hypothesis and telling the world that you have nonsignificant results when: 1. the null is in fact false 2. the predicted relationship does exist in the population 3. the independent does have an effect on the population
Type I Error
-Rejecting the null hypothesis and telling the world that you have significant results when: 1. the null is true 2. the predicted relationship does not really exist in the population 3. the independent variable does not really have an effect in the population *can never decrease it to equal to zero
Alpha
-Symbolizes the "criterion" -The size of the region of rejection of a sampling distribution -If you set alpha = .05, then the size of your region of rejection is .05 *if you are conducting a two-tailed test, the you will have .025 in one tail and .025 in the other tail
When to use a one-tailed test
-When you predict the direction in which scores will change Ex: you predict experimental group will score higher than control group
Why is the critical value smaller for a one-tailed test than for a two-tailed test?
One tailed test- z = 1.645 The region of rejection is concentrated in only one tail - more likely to reject the null and obtain significant results *Means that a one-tailed test has more power than a two-tailed test
Criterion Variable
Outcome variable which may be predicted from one or more predictor variables. It is often the main focus of the study in that it is the outcome variable. Ex: with the previous insurance example, the criterion variable is the amount of insurance sold.
Power
Probability that you will reject the null hypothesis (and conclude that you have significant results)... When there really is an "effect" in the population (the independent variable has an effect on the dependent) Generally, the more powerful your test, the better
If you wish to determine the area under the curve using z scores that have decimal places (as opposed to "whole-number" z scores), what must you do in general? That is, what will you need to use in order to do this?
You will need a table of z in the back of the book
How to convert a raw score into a z score
Z score = X (subject's score) - X_(sample mean) % Sx (sample standard deviation)
How do we show that the results are significant?
p < .05
Different ways to show that results are statistically non-significant?
p > .05 or ns
If I give you the mean and standard deviation for a fictious test, you should be able to compute the upper and lower limits of...
the middle 68% of the class = the middle 96% of the class = the middle 99% of the class =
Region of Nonsignificance
the middle 95% of the sampling distribution.
A sampling distribution of means has a mean that is equal to what?
the population mean that is described in the null hypothesis, or the control population
What percent of area under the curve lies between the mean and...
z = 1.00 34% z = 2.00 48% z = 3.00 49.87% (49%) z = infinity 50%
Two symbols for the standard error of the mean
1. cursive o with a subscript of X with a line 2. SEm
Manipulated Variable
Value of the variable assigned to each subject determined by the researcher
What is one problem associated with the use of point estimates (such as the mean)? What is sampling Error
1. Problem of Accuracy- When there are only 41 people in a sample, we do not expect their mean to be exactly equal to the population mean. Sometimes it will be a little bit higher and sometimes it will be a little lower, just due to sampling error (sampling error is the difference between sample statistics and population parameters due to the fact that samples are not usually perfectly representative of the populations from which they are drawn)
Effect of Transformation from raw into z on the shape of a distribution of z scores
1. Transforming to z scores does not convert a non-normal distribution into a bell-shaped, normal distribution 2. The mean of any distribution is always equal to zero 3. the standard deviation of any z distribution is always equal to one
What are the three experimental characteristics of experimental research
1. Subjects are randomly assigned to experimental conditions 2. The researcher manipulates and independent variable 3. Subjects in different experimental conditions are treated similarly with regard to all variables except the independent varaibel
What are the implications of setting alpha = .05 rather than .01?
.05... Region of Rejection is large Need a small z obt to be significant Power is high only 5% chance of making a type I error .01.... Region of rejection is small Need a big z obt to be significant Power is low Gives you a high level of confidence Type 1 error is VERY small,
What are the disadvantages associated with using a one-tailed test?
1) If obtained z statistic is in the direction you did not predict, your results will be nonsignificant (even if obtained z obt is very large, but negative) 2) If you say you obtained significant results using a one-tailed test many researchers will be suspicious that you only used the one-tailed test... -because your results would not have been significant with a two-tailed test
Two Possible Explanations for the Sample Mean being difference form the mean for the population
1. "Sampling Error" Explanation- the difference is merely due to sampling error, it is NOT due to any real difference. 2. "Real Effect" Explanation- the observed difference between the sample mean versus the populations was NOT due to sampling error.
Confidence Interval
1. Best def. = A range of the values of a population parameter which are not significantly different from the current sample statistic (at a given level of confidence) If you compute a 95% confidence interval, it gives you a range of population parameters that your statistic does not significantly differ from with alpha set at .05 99% = .01, 99.9% = .001
Assumptions underlying the z test for a single sample mean
1. Interval or ration levels scores- the dependent scores that you are analyzing should be on an interval or ratio scale of measurement. 2. Random Sampling- the scores that make up the sample should be randomly selected from the population of interest. 3. Independent Observations- Selecting one subject for inclusion in your sample should have no effect on the probability of selecting any other subject for your sample. Random sampling should satisfy this. 4. Known Population Standard Deviation- The standard deviation of the raw scores in the population must be known 5. The value of the population should not be affected by the treatment/intervention 6. Normally-distributed sampling distribution of means- the sampling distribution of means should display a standard normal distribution (if your original population of raw scores is NOT normally distributed, you can still be confident that this assumption will be met, as long as the number of subjects in your sample is greater than or equal to 30 (central theorem assures this)
Why is the sampling distribution of means almost always a "null hypothesis distribution"?
A distribution that contains the statistics that would be obtained if the Ho were true
Multiple-Item Rating Scale
A group of items which is administered to participants, with responses to those items being either averaged or summed to produce a single score.
Significance Test
A procedure that allows you to test a statistical null hypothesis. It allows you to either reject or fail to reject a statistical null hypothesis with a given level of probability. If you reject the null, your results are statistically significant.
How is a single-sample experiment conducted, in general?
A sample of participants are drawn form a larger population
Critical Value
A score that marks the edge of the region of rejection in a sampling distribution. Obtained sample statistics that are larger than the critical value lie in the region of rejection, and are considered to be statistically significant.
Standard Error of the Mean
A standard deviation of a sampling distribution. A general term, and it is used to refer to the standard deviation of sampling distributions for different types of statistics. The standard error of the mean is the standard deviation of a sampling distribution of means
Statistical Null Hypothesis
A statement that, in the populations being studied, there are (a) no differences between the conditions being studied and/or (b) no relationships between the variables being studied
Point Estimate
A statistic (a single number) obtained from a sample that serves as our "best" estimate of the corresponding population parameter.
Two-Tailed Significance Test
A test in which the region of rejection is divided into two tails of the sampling distribution
Naturally Occurring Variable
A variable which is not manipulated or controlled by the researcher, it is simply measured it as it normally exists (examined in non-experimental research)
Assume that you conduct a z test and the computer gives you a p value of p = .0211. What is the preferred way of presenting your results in a paper, according to the most recent recommendations of the APA>
APA generally prefers we use the new school approach. If you don't have a computer application, you are forced to use the old school method
Percentile Rank
Definition- percent of subjects who scored at or below a given score (Ex: if someone has a percentile rank of 16, it means that he scored higher than 16% of all subjects) z = -2 means that the percentile rank is 2% z = +2 means that the percentile rank is 98%
Single-Item Rating Scale
An instrument in which just one questionnaire item is used to assess some construct of interest. Does not necessarily have equal intervals
An incorrect (although poplar way to interpret confidence intervals)
BAD interpretation = there is a 95% probability that the actual population mean lies somewhere between 20.47 and 23.53 GOOD = We can be 95% confident that this sample confidence interval is one of the sample confidence intervals that captured the true population mean. *The procedures in this course allow us to make probability statements about sample results, but not about population parameters. This applies to the confidence intervals that we compute
Way to summarize a confidence interval in a research article
Because the mean for the sample of 41 vegetarians was 22 minutes, our best estimate for the men for the population of vegetarians is also 22 minutes. The confidence interval around this sample mean was 95% C.I.: 20.47-23.53, which means that the obtained sample mean of 22 minutes was NOT significantly different from all population means that ranged from 20.47 to 23.53 (alpha = .05)
Why are confidence intervals useful?
Because the width of your confidence interval tells you how much confidence you can have in your point estimate. If your confidence level is relatively narrow, then you can be confident that your point estimate (based on the sample) is fairly close to the population parameter. *Narrow confidence intervals are good news
Sampling Distribution of Means
Definition- a theoretical distribution of all possible values of a statistic that are obtained when an infinite number of samples of the same side n are randomly selected from one raw-score population. *Theoretically created by combining an infinite number of sample means in a distribution
Z distribution
Distribution produced by transforming all raw scores in a distribution to z scores
Research Question
Do vegetarians display a mean score on the treadmill minutes test that is significantly different from the mean score displayed by the populations of meat-eaters?
Interval Scales
Does display the property of "equal intervals" with an interval scale, equal differences between scale values do have equal quantitive meaning. Does not have a true zero point (multiple-item summated rating scale) Ex: IQ
Ratio Scales
Have a true zero point- makes it possible to make meaningful statements about the ratios between scale values (ex: Age)
Assume that you are conducting a single-sample z test, that alpha = .05 , and that it is a two-tailed test. How many standard errors must your obtained sample mean be away from the population mean in order to reject the null hypothesis and conclude that you have significant results?
If the sample mean (X_) is more than 1.96 standard errors away from the population mean, you may reject the null hypothesis and conclude that the observed difference is statistically significant. (the middle 95% of a normal distribution will consist of z scores that fall between a z score of -1.96 and +1.96. So 1.96 is the critical value that separates the middle 95% of a distribution from the extreme 5% of the distribution that lies in the two tails.
A sampling distribution of means in NOT a distribution of raw scores,
In contrats, each score in a sampling distribution of means is a mean (an average) based on a sample that contains multiple subjects.
What results must you obtain
In one-tailed test, z obt is significant only if: a) it lies beyond z crit and b) it has the same size as z crit
"Standardized Difference" index of effect size
Indices which indicate how far apart two means are, as measured in population standard deviations (ex: Cohen's d statistic- more popular choice) *Tells us the size of the difference between the sample mean versus the population mean as described under the null hypothesis
"Variance Accounted For" index of effect size
Indices which indicate the percent of variance in the criterion variable which is accounted for by the predictor variable. (ex: r squared, R squared and n (eta) squared
LCL and UCL
Lower Confidence Limit and Upper Confidence Limit
Region of Rejection
Most extreme 5% of sampling distribution (which lies in the two tails). Size of region of rejection corresponds to the "alpha level"
Which type of standard deviation should be used if you wish to convert raw scores into z scores? (For example, should you sue the estimated population standard deviation?)
No, You need the sample standard deviation
Cause-and-Effect Relationships in Non-experimental Research
Non-experimental research that investigates the relationship between just two variables generally provides very weak evidence concerning cause-and-effect relationships. It is often possible to generate a number of different alternative explanations for the same research findings.
When you conduct non-experimental research, you often cannot be confident that your predictor variable really had a casual "effect" on the criterion variable, even when the results are statistically significant. Why?
Non-experimental studies often suffer form confoundings and poor experimental control
Effect Size
Refers to the strength of the relationship between a predictor variable and a criterion variable in an investigation (experiment = the strength of the effect that the independent variable had on the dependent, correlational = the strength of the association between the predictor variable and the criterion variable)
One-Tailed Significance Test (also called a directional test)
Region of rejection falls in only one tail of the sampling distribution
What makes it possible for researchers to use the standard normal curve to determine the relative frequency of a z score?
Relative frequency is the same things as the total area under the curve because there is a precise relationship between z scores and area under the curve (almost all scores will lie between -3.00 and +3.00)
Dependent Variable
Some aspect of the subject's behavior which is assessed to determine whether it has been affected by the independent variable. The experimental counterpart to a criterion valuable. (depends on the independent)
In a single-sample experiment, you wish to determine whether the observed sample mean is significantly different from?
Some population mean or theoretical mean
Statistical Null Hypothesis
Statement that, in the populations being studied, there are (a) no differences being studied, and/or (b) no relationships between the variables being studied *Researchers always hope to reject the null hypothesis
Central Limit Theorem
States that a sampling distribution of means will have a certain predictable characteristics. It is probably the single most important theorem in a stats course. IT states that a sampling distribution of means will have the following characteristics... 1. It will form an approximately normal distribution, even if the distribution of raw scores on which it is based is not normally distributed 2. It will have a mean that is equal to the mean of the raw-score population 3. It will have a standard deviation (standard error) that is mathematically related to the standard deviation of the raw scores
Nondirectional Research Hypothesis (also called "two tailed" hypothesis)
States that the sample will display a mean score that is different from the population, but it does not state whether the mean will be higher or lower than the mean for the population.
Region of Rejection
That portion of a sampling distribution containing values considered too unlikely to occur by chance, found in the tail or tails of the distribution
What is sampling error?
The difference (due to change) between: 1. The obtained sample statistic 2. The corresponding population parameter * One word synonym used for sampling error = chance
Nonexperimental (correlational) Research
The researcher simply studies the naturally-occuring relationship between two naturally-occuring variables
Alpha Level
The size of the region of rejection for the significance test 1. If you set alpha at .10, the the region of rejection consists of the most extreme 10% of the distribution, out in the tail or tails 2. If you set alpha at .05, then the region of rejection consists of the most extreme 5% of the distribution, out in the tail or tails 3. If you set alpha at .01, then the region of rejection consists of the most extreme 1% of the distribution, out in the tail or tails 4. If you set alpha at .001, the the region of rejection consists of the most extreme .1% of the distribution, out in the tail or tails
Which should be smaller, standard error of the hand or standard deviation?
The standard error of the mean is smaller that the standard deviation of the raw score population.
Critical Value
The value of the sample statistic that marks the edge of the region of rejection in a sampling distribution -Values that fall beyond it fall in the region of rejection -with a z-test, symbol is zcrit
Independent Variable
The variable whose values (or levels) are selected by the experimenter to determine what effect the independent variable has on the dependent variable. The variable that is actively manipulated by the researcher, experimental counterpart to a predictor variable. (stays the same)
Why does sampling error occur, in general?
This difference is due to the fact that samples are often not perfectly representative of the population from which they are drawn. The mean for the sample is not identical to the mean for population because the sample was not perfectly representative of the population from which it came.
What type of data may z scores be computed from?
To compute z scores, the raw scores should be on an: interval scale or ration scale
Why is the term single-sample experiment misleading?
Two ***** 1. The term "single sample" may mislead readers into assuming that there is only one condition in a single sample investigation. In fact, there are two conditions (sort of): a. experimental condition- the condition that is experienced by the sample of interest- separates them from the normal population b. control condition- the population of so-called normal individuals 2. Not a true experiment- researchers generally think of an experiment as being a special type of investigation that involves random assignment to conditions, manipulation of a true independent variable, and a high degree of experimental control. It is relatively unusual that a single-sample investigation displays such high standards of experimental rigor.
Statistical Alternative Hypothesis
Typically a prediction that there is a difference between groups in the population, or there is relationship between variables in the population. It is the counterpart to the null hypothesis if you reject the null hypothesis, you tentatively accept the alternative hypothesis Ex: In the population, there is a difference between the high goal-difficulty and the low goal-difficulty group with respect to their mean scores on the amount of insurance sold.
New School Approach
Uses the computer application SSPS or SAS "My results were statistically significant, z = +2.01, p = .0444" "My results were statistically nonsignificant, z = +1.27, p = .2040
Predictor Variable
Variable used to predict values on the criterion. In some studies, you may even believe that the predictor variable has a causal effect on the criterion. Ex: Because you believed that goal difficulty may positively affect insurance sales, you conducted a study in which goal difficulty was the predictor and insurance sales was the criterion.
Ordinal Scales
Variables on this scale represent the rank order of the subjects with respect to the variable being assessed. "Ranking variables (ex: runners in a race)
Z Scores
What are they? Number that indicates the distance that a raw score deviates from the same mean in standard deviations What do they tell us? 1) Where a subject stands in relation to the mean (+ = subject's score is above the mean, - = subject's raw score is below the mean, 0 = subject's raw score is the mean) 2) How far a subject's raw score deviates from the mean (most will be between +3.00 and -3.00)
Formular for Cohen's d statistic
d = X_ - delta % cursive o X_ = obtained sample mean delta = theoretical/population mean cursive o = standard deviation of the raw score (control) population *not standard error of the mean
The ____ of a ______ of subjects is usually your best estimate for the mean of a population
mean, representative sample
According to Cohen's guidelines for interpreting his d statistic, how large should d be to consider it a small, medium and large effect>
small d = .20 medium d = .50 large d = .80
