Psych Stats Exam 2

APA style write-up

-People using 42-in monitors (Xbar = 116 seconds) complete tasks faster on average than people using 14-inch monitors (Xbar = 127 seconds), t(4) = -5.73, p < .05. -There was a significant difference in number of words recalled, such that the Caffeine Group (Xbar = 12.67) recalled significantly more words than the No Caffeine Group (Xbar = 10.33), t(10) = 2.57, p < .05. The effect size is very large d = 1.49 and the 95% CI is [0.31, 4.37]. -There is no statistically significant difference in anxiety scores between the general population of people with clinical anxiety (µ = 8.00) and people with clinical anxiety who complete my treatment program (M = 12.20), t(9) = 1.86, p > .05.

3 ways to determine statistical significance

1. Compare t statistic from the experiment to the critical t value Reject the null if: absolute value of observed t is greater than critical t 2. Look at the p value and compare it to alpha Reject the null if: p value is less than set alpha level 3. Check whether the confidence interval includes mean of the sampling distribution Reject the null if: CI doesn't include mean of the sampling distribution

General Steps of NHST

1. Specify null and alternative hypotheses 2. Assume the null hypothesis is true 3. Choose a sampling distribution 4. Choose a value to set critical value and rejection regions: alpha 5. Collect sample(s) and calculate mean(s) 6. Calculate t statistic 7. Make a decision

Effect Size

Cohen's d is a measure of effect size that tells us how much two means differ from each other in population standard deviation units. Effect size is a descriptive statistic, so it's about the experiment -Less affected by sample size -Affected by difference between means -Affected by variability in population

1. Specify null and alternative hypotheses

Null hypothesis: There is no difference Alternative hypothesis: There is a difference One-sample t: Null: (μ0 = μ1), Alternative: (μ0 ≠ μ1) Paired-samples t: Null: (μ1 = μ2), Alternative: (μ1 ≠ μ2) Independent-samples t: Null: (μ1 = μ2), Alternative: (μ1 ≠ μ2) These are two-tailed alternative hypotheses. We can also specify the direction of the difference with a one-tailed alternative hypothesis. How does a one-tailed test affect NHST? Smaller critical t, so we can more easily reject the null hypothesis How does it affect power? Increases power but we miss the effect if it's in the other direction

3. Choose a sampling distribution cont.

One-sample t: SAMPLING DISTRIBUTION OF MEANS Paired-samples t: SAMPLING DISTRIBUTION OF MEAN DIFFERENCES Paired-samples t: SAMPLING DISTRIBUTION OF MEAN DIFFERENCES

Biased sample

People are picked based on availability, conclusion's generalizability is questionable, replication helps overcome drawbacks of this method

Unbiased sample

Random selection, each person has an equal chance of being selected, sample is representative of population, ideal but rare method

Power

Ability to reject the null, given that the null is really false. Probability of finding an effect, if it exists. Probability that we avoid a Type II Error.

paired samples t-test

Two samples; Within-groups design OR between-groups design with matched or natural pairs

4. Choose a value to set critical value and rejection regions: alpha

Use t table here Check whether one-tailed or two-tailed test and look at correct alpha level on the table Calculate degrees of freedom One-sample t: N - 1 Paired-samples t: N - 1 Independent-samples t: (N1 - 1) + (N2 - 1)

Power cont.

We typically set alpha to 0.05, but let's say we set it to 0.02. What happens to power, Type 1 error and Type 2 error? Type 1 Error DECREASES Type 2 Error INCREASES Power DECREASES

t distrubution

We use the t distribution as our sampling distribution Looks normal-ish Used when we don't know sigma Multiple t distributions identified by degrees of freedom Becomes normal as the degrees of freedom increase

1) Use a higher alpha level (e.g., 0.1 instead of 0.05)

What's wrong with this approach? Increasing alpha level also increases the probability of a Type I error. This is not usually a good method for increasing statistical power.

7. Make a decision

Where does your observed t statistic fall in the sampling distribution? Reject the null if observed t falls in the rejection regions. Retain the null otherwise.

2. Assume the null hypothesis is true

this lol

Five factors that increase power

1) Use a higher alpha level (e.g., 0.1 instead of 0.05) 2) Use a one-tailed hypothesis instead of a two-tailed hypothesis 3) Increase sample size 4) Reduce variability 5) Make the difference between population means bigger

Sampling distribution definiiton

A theoretical frequency distribution of sample statistics (usually the mean) from a bunch of samples drawn from the same population. Almost always normally distributed, even if underlying population is skewed Has same center as population distribution Narrower than population distribution (standard error < sigma), gets narrower as N of each sample increases More closely approximates a normal curve as N of each sample increases

2) Use a one-tailed hypothesis instead of a two-tailed hypothesis

A two-tailed test divides alpha into two tails. When we use a one-tailed test, putting the entire alpha into just one tail, we increase the chances of rejecting the null hypothesis, which increases statistical power. What's wrong with this approach? We have no chance of detecting an effect if it is in the opposite direction. What type of error is this? Type II error!

3. Choose a sampling distribution

All statistical tests in this unit are t tests, so each sampling distribution is t distributed.

3) Increase sample size 4) Reduce Variability

As sample size increases, the distributions of means become narrower and therefore provide more power. The same is true for decreasing variability (standard deviations). What might be wrong with this approach? Increasing sample size allows us to detect trivially small differences. We can rely on effect size to determine if these differences are meaningful!

5) Make the difference between population means bigger

As the difference between means becomes larger, there is less overlap between curves. This translates to higher statistical power

Confidence Interval

CI is an interval estimate that provides a range of possible values for the population parameter. CIs give us a measure of precision for our estimate. Narrower intervals are better. A 95% CI means: If we repeatedly sampled from the population, the interval would contain the point estimate 95% of the time. (Point estimate) ± (Critical t from table)*(SD of sampling distribution) CI: [lower limit, upper limit]

Type 1 and Type 2 Errors, Power

Let's assume α = 0.02. What will happen to Type 1 and Type 2 errors and Power? Type 1: will decrease Type 2: will increase Power: will decrease

Sampling distribution

Mue x bar (expected value) and Sigma x bar (standard error)

single sample t test

One sample, only mue is known do not know sigma

p value

The p value is the probability of observing our data (or more extreme data) given that the null hypothesis is true. The p value is NOT the probability that our data are due to chance. The p value is NOT the probability that the null hypothesis is true. The p value is NOT the probability of a Type I error. The p value is NOT the probability of making a wrong decision. 1-p is NOT the probability that the alternative hypothesis is true.

Central Limit Theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution.

independent samples t-test

Two samples; Between-groups design

Sample

X bar, S

5. Collect sample(s) and calculate mean(s)

do the math

Population

mue, sigma, s hat