Hypothesis Testing

The risk of a Type _ error is more serious than a risk of a Type _ error.

Type I; Type II- Rejecting a true null hypothesis which is a Type I error is considered the more serious of the two possible errors

What does the Z score tell us?

-How far the SD corresponding value of x (score) lies above/below the mean. -This standard unit of measure allows us to make comparisons between different variables.

Constructing a confidence interval:

-critical value is chosen -Because distribution of sample means is assumed to be normally distributed, 95% of sample means are expected to fall within 2 standard errors of population mean. -Thus, critical value for a 95% confidence interval is equal to 2. -The critical value depends on level of confidence chosen by researcher.

Interval Estimation

-the establishment of a range of values that we can say with confidence contain the population parameter -range of values is the confidence interval

Alpha is conventionally set at a conservative level of __________.

.05 or less

What are the 3 important characteristics of normal distribution?

1) it is unimodal (one mode at center) and symmetrical 2) it is continuous 3) it is asymptotic: curved line gets closer to horizontal axis as it moves away from the center

When distribution of sample means is known, it is easy to compute probabilities. Typically not practical because target population is large. So, we use a theorem:

1) mean of the distribution of sample means equals the mean of the population 2) distribution of sample means is less variable than the population: -This means that the standard deviation of the sample means is smaller than the standard deviation of the population - standard deviation of sample means is known as the standard error of the mean or SEM -to determine this, we need to have the parameters of the population which we typically do not have available - So: we use a solution, called the Central Limit Theorem

What is the problem with the one-tailed alternative hypothesis?

1) one-tailed ignores possibility of unexpected difference in opposite direction; 2) more vulnerable to error if distribution is not normal

Two types of alternative hypothesis:

1) one-tailed or directional: predicts that one treatment or method will be better than another. 2) two-tailed: does not predict direction; just hypothesizing that the two treatments or methods are significantly different

To make reliable decisions about research hypotheses, researchers consider 2 opposing points of view:

1. Null hypothesis: in testing a statistical hypothesis, what is expected when the hypothesis is TRUE must be known. Thus, researchers hypothesize the OPPOSITE of what they expect; thus, researchers hypothesize that there is NO difference between two methods or two groups, etc. (NULL); usually stated as no relationship between variables or no difference between groups. Ex: the world is flat 2. Alternative hypothesis: This is a statement of the expected result of study- that there will be a significant difference between the groups, methods, etc. relative to the specific variable examined Ex: the world is round

What are the steps to take when testing the hypothesis?

1. Stating the hypothesis 2. Set an acceptable level of risk, referred to as the alpha level. 3. Choosing the sample size. 4. Determine the critical value. 5. Compute the test statistic 6. Make a Decision about Null Hypothesis.

Z scores are transformed into a distribution of standard scores with a mean of ___ and SD equal to ___ units.

100; 15

On a bell-shaped curve, Area between -1 and +1 standard deviations always includes: Area between -2 and +2 SD includes: Area between -3 and +3 SD includes:

68% of total distribution of scores 95% of scores 99% of scores **This information is important for determining the probability of an outcome.

All normal distributions can be converted to what?

A common distribution with the same mean and standard deviation (mean of zero and standard deviation of one). *THIS IS THE Z STATISTIC

What is a distribution and what does it provide information about?

A distribution is a pattern of scores; distribution of a variable provides information about individual cases as well as information about the group of scores.

When Test statistic or Observed Value (finding from running the statistical test) is less than data value for Critical value (cut-off), null hypothesis is:

ACCEPTED

Testing the hypothesis: Step 6 (Make a Decision about Null Hypothesis)

After test statistic (Observed Value) is computed, it is compared to the Critical Value on the table. If test statistic (Observed Value) is greater than the Critical Value, null hypothesis is rejected. If this is the case, alternative hypothesis is supported/accepted but not proven. If test statistic/Observed Value is less than Critical Value, null hypothesis is accepted.

Z statistic

Critical Value is a standard score; examining the standard normal distribution

Power

Defined as the probability of rejecting the null hypothesis when it is false which would be a correct decision.

Rejection region

Dependent upon the critical value. Rejection region may be located in either tail depending upon whether one or two-tailed alternative hypothesis. One tailed is more powerful analysis because rejection region is larger

Testing the hypothesis: Step 1 (Stating the hypothesis)

Descriptive vs. inferential statistics Descriptive Statistics: organize, summarize, describe data WITHOUT making inferences about the population. Hypothesis testing is an important part of Inferential statistics: make conclusions about populations of individuals from sample data - making an INFERENCE

Distributions for categorical variable are shown in ______________. Distributions for continuous variables are usually displayed in ___________ or ______________.

Distributions for categorical variables are shown in bar graphs; Distributions for continuous variables are usually displayed in line graphs or histograms

The distribution of a sample statistic indicates:

How often different values of that statistic should occur if samples of the same size are collected repeatedly from the same populations. One example: distribution of sample means (for a particular variable)

What kind of relationship is between a alpha and a beta?

Inverse relationship- with other factors constant, an increase in alpha is accompanied by a decrease in beta or a decrease in alpha is accompanied by an increase in beta.

When distribution of sample means is known, ____

It it easy to compute probabilities. *Typically not practical because target population is large.

Kurtosis

Measure of peakedness for symmetrical distributions; measure of how fat or thin the tails of a distribution are relative to a normal distribution

The central limit theorem says that a sampling distribution of sufficient size is approximately _______.

Normal

Skewed Distributions

Not bell-shaped and not symmetrical.

The sample size determines what two things?

Probability of distribtion to be used & the power

Probability distributions

Provides information about the chance occurrence of a particular outcome for a particular distribution of scores

When Test statistic or Observed Value (finding from running the statistical test) is greater than data value for Critical value (cut-off), null hypothesis is

REJECTED and alternative hypothesis is ACCEPTED

In regards to CSD, why do researchers often depend on Student's t distributions to test hypotheses?

Research in CSD often involve small samples.

Testing the hypothesis: Step 3 (Choosing the sample size)

Sample Size is n

What is the major factor that influences power of a statistical test?

Sample size; So, a larger sample usually results in a more powerful test of the null hypothesis.

Why is the possibility of committing an error so important in hypothesis testing?

Science demands a high degree of reliability (REPEATABILITY)! So, minimizing risk of errors is very important in the hypothesis testing process.

Hypothesis

Statements that describe the proposed relationship between two or more variables.

Testing the hypothesis: Step 4 (Determine the critical value)

Statistical hypothesis testing requires a cut-off point that can be used to separate sample results that lead to rejecting the null hypothesis from results that lead to accepting the null hypothesis.

Testing the hypothesis: Step 5 (Compute the test statistic)

Test statistic: statistical test you will use to analyze your data

What does the Critical value depend on?

The alpha level (level at which agree to accept/reject null hypothesis) and alternative hypothesis

If your alpha level =.05 (p value), what does this mean?

The researcher is willing to risk a Type I error 5% of the time or 5 out of 100 times;

Single score in distribution of scores in standard normal distribution can be represented by standard unit of measure:

The z score

Alternative hypotheses should be two-tailed unless:

There is a compelling reason or evidence that result will be directional. Directionality should be decided before beginning of study.

Why are z scores difficult to report or may be misleading?

They include negative values and decimals.

Testing hypotheses is a binary decision-making process, meaning what?

This means that the examiner will either accept or reject hypotheses based on the results of statistical tests.

Testing the hypothesis: Step 2 (Set an acceptable level of risk, referred to as the alpha level)

When testing a research hypothesis, 4 possible outcomes or decisions: 1) null hypothesis is accepted when it is true (correct decision); 2) null hypothesis is rejected when it is false (correct decision); accepting alternative hypothesis 3) null hypothesis is rejected when in reality it is true (Type I error); accepting the alternative. 4) null hypothesis is accepted when in reality it is false (Type II error)

t- distributions

distributions are symmetrical , bell-shaped, and centered on the mean; however, distributions change as sample sizes change

Choice of statistic depends upon:

hypothesis distribution of sample population sample size other sample characteristics

Probability of Type II error:

is designated by beta, the value of which cannot be determined prior to study unless parameters of population are known. Thus, beta is rarely specified because research in CSD usually involves populations with unknown parameters.

Probability of Type I error:

is designated by the Greek letter, alpha, which is specified by the researcher before the data collection begins

__________ skewed distribution has more scores with larger values toward the right tail, whereas a __________ skewed distribution has more scores with larger values toward the left tail

negatively; positively

What test may be more appropriate if the population is not normally distributed?

nonparametric tests may be more appropriate because they are distribution-free - these tests do not depend on the population conforming to a prescribed shape because raw data is converted to ranks (ordinal form).

A result may be statistically significant but may not be a _______ _________.

practical value; a statistically significant result may not be clinically relevant.

If characteristics of a target population are unknown, population parameters can be estimated from _______________.

sample parameters

Clinical significance is determined by:

several factors but the most important is effect size (ES) * helps establish evidence-based clinical practices

z score=

standard score; It is a measure of individual location, telling us where individual scores are located within a distribution of scores.

If hypothesis is examining a difference between two sample means (groups), which test would be appropriate?

t-test

What is the cut-off point?

the Critical Value which can be found using a table for the standard normal distribution (z statistic) or the Student t distribution.

What is the standard error of the mean estimated from?

the SD of the sample

All normal distributions are transformed to fit:

the standard normal distribution

What is the goal of hypothesis testing?

use collected data from samples to decide whether to accept or reject the hypothesis.

What happens when we draw conclusions from studies with small samples (n<10)?

we are at higher risk for accepting the null hypothesis when it is false (Type II error) - low on POWER.