Stats test 2 study guide

Normal distributions

Symmetrical, mesokurtic, unimodal, y-axis is probability distribution, area under curve = probability, total area under curve = sample space

x-axis = continuous variable

Model distributions

If you can assume the population is similar to the model, you can simulate population data with this model.

The “68-95-99” rule

68% of the data will be within +- 1 standard deviations from the mean

95% of the data will be within +- 2 standard deviations from the mean

99% of the data will be within +- 3 standard deviations from the mean

Z score for pop mean

(standard score) indicates how many standard deviations a data point is from the mean, on the x-axis.

Formula: Z=x-µ/σ

Standardization to Z

Re-expressing data using a common standard. Standard normal distribution has a mean of 0 and a sd of 1.

Central limit theorem

The averages of samples have approximately normal distribution Sample size→ bigger distribution→ more normal and narrow.

Sampling error

Sample statistics estimate population parameters with varying degrees of success

-efficiency and resistance

-Sampling error refers to the deviation between a statistic and its parameter

Sampling distributions

obtain an infinite number of samples of size=n from population

calculate statistic for each sample

plot a freq distribution for each sample

3 characteristics of sampling distribution of the mean

Given Pop mean= µ and variance= o²

Mean = µ,

variance = σ²/n

As N goes up, the shape of the distribution approaches normality

Standard error

Variance= σ²/n,

Formula is sd or σ= σ/√n

Z score for sample mean

Z= x̄-µ / σ/√n

Hypothesis testing

• State the Hypotheses: * $H_0$ (Null): There is no effect or difference.

  • $H_1$ (Alternative): There is an effect/difference.

• Set the Decision Criteria: Choose an alpha level ($\alpha$), typically .05. This defines the "critical region."

• Calculate the Test Statistic: For a Z-test: $z = \frac{\bar{M} - \mu}{SE}$.

• Make a Decision: Compare your calculated $z$ to the critical $z$ (e.g., $\pm 1.96$ for $\alpha = .05$). If your $z$ is in the critical region, Reject $H_0$.

• State the Conclusion: Describe what the results mean in the context of the original research question.

• Type I Error

• Type II Error

• Type 1: A "False Positive." Rejecting the null hypothesis when it is actually true (saying there is an effect when there isn't).\

• Type II Error: A "False Negative." Failing to reject the null hypothesis when it is actually false (missing a real effect).

Cohen’s D

7. Effect Size (Cohen’s D)

While a Z-test tells you if a result is significant, Effect Size tells you how much it matters. It measures the magnitude of the treatment effect independent of sample size.

• Cohen’s D Formula: $d = \frac{\bar{M} - \mu}{\sigma}$

• General Guide: $0.2$ = Small; $0.5$ = Medium; $0.8$ = Large.

Statistical power

Statistical Power

Power ($1 - \beta$) is the probability that a test will correctly reject a false null hypothesis. In short, it’s the ability of your study to detect an effect if one truly exists.

Increasing power

1. Increase Sample Size ($n$)

This is the most direct way to boost power. A larger $n$ reduces the Standard Error ($SE = \sigma / \sqrt{n}$), which "shrinks" the sampling distributions. This creates less overlap between the Null ($H_0$) and Alternative ($H_1$) distributions, making it easier to reject the null.

2. Increase Alpha ($\alpha$)

If you move your significance level from $.01$ to $.05$ (or $.05$ to $.10$), you are effectively moving the goalposts closer. By expanding the Critical Region, you make it easier to reject the null hypothesis.

• The Trade-off: While this increases power, it also increases the risk of a Type I Error (False Positive).

3. Use a One-Tailed Test (Directional)

If your hypothesis is directional (e.g., "Group A will score higher than Group B"), you can put all your alpha into one tail rather than splitting it between two. This moves the critical value closer to the mean, requiring a smaller observed effect to achieve significance.

4. Increase the Effect Size

This involves making the "signal" stronger relative to the "noise." You can do this by:

• Using a stronger treatment (e.g., a higher dose of a medication).

• Reducing measurement error by using more precise equipment or highly standardized procedures.

5. Decrease Population Variance ($\sigma$)

Power increases when the data is less "spread out." You can achieve this by using a more homogeneous sample.

• Example: If you are testing a new teaching method, testing only "5th-grade students at one specific school" will likely have less variance than testing "all K-12 students in the country." Lower variance makes the distribution curves thinner, reducing overlap and increasing power.