Psych Stats Test 2

The likelihood of an event occurring

= specified outcome / total outcomes

12:52 or 3:13

0.23

23.08%

Used to calculate the likelihood of obtaining a specific sample from a given population

If the probability of getting a specific the sample is low, we can say that the sample probably came from some other population

Random sampling is a statistical technique used to select a subset of individuals or items from a larger population. It involves selecting individuals or items in such a way that each member of the population has an equal chance of being chosen. This method helps to ensure that the sample is representative of the population and reduces bias.

• Items can be selected from a population multiple times

• Each time an item is selected, it is returned to the population before the next selection is made.

• Commonly used in probability theory and statistics to simulate random processes and estimate population parameters.

• each member of the population has an equal chance of being chosen no matter what has already been selected

• Items can be selected from a population only once

• Each time an item is selected, it is removed from the population before the next selection is made.

• Commonly used in probability theory and statistics to ensure each selection is independent

• Each member of the population’s chance of being selected increases after selections have been made

• Draw a vertical line at the data point

• The line divides the distribution into 2 sections: the body and the tail

• The exact location of the line can be specified by a z-score

• Look at z-score table to see the proportion in body and tail.

First, transform the score into a z-score

Then look up the z-score in the table and read across the row to find the appropriate probability

First, look up the proportion in the table and read across the row to find the appropriate z-score

Then transform the z-score into a x-value

-+1.96

Indicates the percentage of data points that are equal to or below a specific value in a dataset

For example, if a student scores in the 80th percentile on a standardized test, it means they performed better than 80% of the test-takers.

To calculate the percentile, follow these steps:

1. Arrange the data in ascending order.

2. Determine the proportion of scores equal to or less than the score.

A probability distribution that is symmetric and bell-shaped. It is characterized by its mean (μ) and standard deviation (σ).

• The majority of the data falls near the mean, with fewer data points further away from the mean.

• The shape of the distribution is determined by the mean and standard deviation.

• The area under the curve represents the probability of observing a particular value or range of values.

The difference between a sample statistic and the true population parameter it represents.

It occurs due to the inherent variability in the process of selecting a sample from a larger population.

Quantified using measures such as margin of error or standard error.

Provides a measure of the average expected distance between M and mu

The larger the sample size (n) in a specific sample the more probable the M is close to mu

(the larger the sample size the small the standard error of the mean)

The bigger the sigma the wider the distribution

the population mean

• The shape of the distribution of sample means is typically normal

• Distribution of sample means approaches a normal distribution as n approaches infinity

the larger the sample size the Smaller the variability

• the population the samples are obtained from is normal

• the sample size is n=30 or more

Statistical method that uses sample data to evaluate a hypothesis about a population

To rule out chance (sampling error) as a plausible explanation for the results from a research study

The observed findings are due to random chance (there does not appear to be a real effect)

Predicts that the independent variable had no effect on the dependent variable

The observed findings cannot be explained by sampling error (there does appear to be a real effect)

Predicts that the independent variable did have an effect on the dependent variable

H0

H1

1. State your hypothesis about the population

2. Define the probability at which we think results indicate that there must be a true effect

3. Obtain and test a sample from the population

4. Compare data with the hypothesis predictions, make conclusion

Consists of outcomes very unlikely to occur if the null hypothesis is true

Defined by associations that are very unlikely to obtain (typically less than 5% chance) if no effect exists

Establishes a criterion or “cut-off”, for deciding if the null hypothesis is correct

Typically α = .05 (rarely α = .10 or α = .01)

Forms a ratio comparing the difference between the M and μ versus the amount of difference we would expect without any treatment effect (σM)

If the test statistic results are in the critical region, we conclude the difference is significant (an effect exists)

If the test statistic is not in the critical region, conclude that the difference is not significant (any difference is just due to chance)

There are several factors that can influence a hypothesis test:

1. Sample size: A larger sample size generally provides more reliable results and increases the power of the test.

2. Significance level: The chosen level of significance (e.g., 0.05) determines the likelihood of rejecting the null hypothesis.

3. Effect size: The magnitude of the difference or relationship being tested can influence the outcome of the test.

4. Variability: The amount of variability in the data can affect the precision of the test and the ability to detect significant differences.

5. Assumptions: Violations of assumptions, such as normality or independence, can impact the validity of the test.

6. Type of test: The specific hypothesis test used (e.g., t-test, ANOVA, chi-square) will depend on the research question and data type.

These factors should be carefully considered when conducting a hypothesis test to ensure accurate and meaningful results.

Occur when the sample data indicate an effect when no effect actually exists.

• Rejecting the null hypothesis when the null is true.

• Caused by unusual, unrepresentative samples, falling in the critical region without any true effect.

• Hypothesis tests are structured to make Type I errors unlikely.

Occur when the hypothesis test does not indicate an effect but in reality an effect does exist.

• We fail to reject the null hypothesis even though it was actually false.

• More likely with a small treatment effect or poor study design (sample size too small).

includes a directional prediction in the statement of the hypotheses and in the location of the critical region

Rarely used as it doesn’t make sense

All probability located in one tail, making that tail larger

The probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis

Used when performing hypothesis tests; the statistical significance is checked by seeing if our test scores (ex z-score) indicate a p-value of less than our α

Probably about 20-50% depending on context

Assuming the null is true, you’d obtain the observed difference or more in 5% of studies due to random sampling error.

Measures of the absolute magnitude of an effect, independent of sample size

• Standardized effect size

• Like a z-test, measures mean difference in terms of the standard deviation

Because effect size provides a measure of the practical significance of the results.

• Hypothesis tests determine if there is a statistically significant difference, and effect size quantifies the magnitude of the difference.

Sample Size: Effect size is independent of sample size

Standard Deviation: Effect size is inversely related to the standard deviation

The probability that the test will reject the null hypothesis when there is actually an effect

It represents the probability of detecting a true effect or relationship between variables.

A higher power indicates a greater likelihood of correctly rejecting the null hypothesis and avoiding a Type II error (false negative).

A well-powered test increases the confidence in the results and enhances the reliability of scientific conclusions.

• effect size (larger effects are easier to find)

• sample size (larger samples make it easier to find effects)

• alpha level (larger alpha level makes it easier to find effects

• non-directional vs directional hypothesis (directional tests make it easier to find effects)

"estimated z-test.“

• Estimated because we are using the sample standard deviation to estimate the unknown population standard deviation.

Allows researchers to use sample data to test hypotheses about the difference between a sample mean and a population mean.

The t statistic does not require knowledge of the population standard deviation (σ)

Can be used for a completely unknown population (both μ and σ are unknown)

• All that required is a sample and a reasonable hypothesis about μ

1. One-sample t-test

2. Independent Samples or Independent Measures t-test

3. Dependent Samples or Repeated Measures t-test

How much difference between M and μ is reasonable to expect

“The Noise”

The hypothesis test with a t statistic follows the same four-step procedure that was used with z-tests:

1. State the hypotheses and select a value for α. (Note: The null always states a value for μ.)

2. Locate the critical region. (Note: You must find the value for df and use the t distribution table.)

3. Calculate the test statistic.

4. Make a decision. (Either "reject" or "fail to reject" the null hypothesis.)

Need to calculate the degrees of freedom

• df = n – 1 = 25 – 1 = 24

Then we go to the t distribution table.

• Since we are using a non-directional hypothesis the test is two-tailed

• Since our alpha level is .05 go to column with .05 in two tails

• Go to the row corresponding to our df.

• Take note of Critical Region

• With large samples, the t value will be very similar to a z-test.

• With small samples, however, the t-value will provide a relatively poor estimate of z.

Sample variance affects the t-test by influencing the calculation of the t-statistic

A larger sample variance leads to a larger standard error and a smaller t-statistic, making it less likely to reject the null hypothesis

A smaller sample variance leads to a smaller standard error and a larger t-statistic, making it more likely to reject the null hypothesis.

For t-test

Percentage of Variance Accounted for by the IV

• Scores differ across individuals for many reasons.

• By measuring the amount of variability that can be attributed to the IV, we obtain a new measure of effect size

Used in situations where a researcher has no prior knowledge about either of the two populations (or treatments) being compared.

Allows evaluation of the mean difference between two unknown populations using data from two samples.

Independent-measures designs uses two separate and independent samples.

• Use #1: Test for mean differences between two distinct populations (those with college degrees and those without).

• Use #2: Test for mean differences between two different conditions (Acceptance and Commitment Therapy vs. placebo).

Measured in the same way that we measured effect size for the one-sample t-test

Specifically, can calculate Cohen’s d or r2

1. The data are continuous.

2. The sample data have been randomly sampled from the population.

3. The variability of the data in each group is similar.

• For Independent-Measures t-Tests this means the variances of the populations that samples are drawn from are similar.

4. The sampled population is approximately normally distributed.