Explain the difference between ratio, interval, ordinal, and nominal scales of measurement.

• Nominal: Categories without any order (e.g., gender, eye color).

• Ordinal: Ordered categories with no consistent differences between values (e.g., rankings).

• Interval: Ordered categories with consistent intervals but no true zero (e.g., temperature in Celsius).

• Ratio: Like interval, but with a true zero point (e.g., weight, height).

Explain how skew and kurtosis affect measurements of central tendency and/or measures of variability.

• Skew: Affects the symmetry of the distribution, shifting the mean away from the median.

• Kurtosis: Refers to the 'tailedness' of the distribution, influencing the variability in extreme values.

What is variance? What is the difference between a standard deviation and a standard error of the mean? Explain what a sampling distribution is in your answer.

• Variance: Measure of the average squared differences from the mean.

• Standard Deviation: Square root of variance, indicating data spread.

• Standard Error: Measures the variability of the sample mean relative to the population mean.

• Sampling Distribution: The distribution of sample means over repeated sampling from the population.

Why does sample variance underestimate the true population variance?

• Sample variance underestimates because it divides by N instead of N-1, which leads to missing some variability in small samples.

Explain why N-1 is used as the denominator for sample variance.

• N-1 corrects for bias in estimating the population variance from a sample, known as Bessel’s correction.

Explain what is meant by “efficient,” “unbiased,” “sufficient,” and “resistant” when referring to estimators.

• Efficient: Smallest variance among unbiased estimators.

• Unbiased: Expected value equals the true population parameter.

• Sufficient: Uses all the data to estimate the parameter.

• Resistant: Not influenced by outliers.

How does sample size affect efficiency? What about the standard error of the mean?

• Larger sample sizes increase efficiency and reduce the standard error of the mean.

Why can you use the normal distribution (z) to estimate a binomial distribution? Why should you not use the normal distribution to estimate the binomial distribution when you have small sample sizes?

• You can use the normal distribution for binomial data when sample size is large due to the Central Limit Theorem.

• You should not use it for small samples as binomial data may not be symmetric.

What is a null hypothesis? Explain the logic of null hypothesis significance testing.

• The null hypothesis states there is no effect or difference. Null hypothesis significance testing checks if data provide enough evidence to reject it.

What is a p-value? How do p-values differ from effect sizes? Do sample sizes affect p-values? What about effect sizes?

• P-value: Probability of observing the data assuming the null hypothesis is true.

• Effect Size: Measures the magnitude of a difference.

• Sample size affects p-values but not effect sizes.

What is the purpose of standardizing a distribution using the z-formula? What do the mean and standard deviation become when you standardize a distribution?

• Standardizing allows comparisons across different distributions.

• The mean becomes 0, and the standard deviation becomes 1.

Why do data need to be normally distributed when conducting a z-test or a t-test?

• Z-tests and t-tests assume normality for accurate p-values and test validity.

Explain the similarities and/or differences between the z and t distributions. In what instance are the z and t distributions the same?

• Both are similar, but the t-distribution has heavier tails, used for smaller sample sizes.

• They are the same when the sample size is large (infinite degrees of freedom).

Why do the critical values for a t-test change? Unlike z-tests, where the critical cut-offs are either 1.64 or 1.96 at an alpha of 0.05.

• T-distribution critical values change with sample size due to the increased uncertainty in small samples.

What is the difference between a one-sample t-test and a z-test? If you are in a situation where either test is possible to conduct, which would you choose and why?

• T-test: Used when population standard deviation is unknown.

• Z-test: Used when population standard deviation is known.

• Choose the t-test if population variance is unknown.

What are the assumptions for a one-sample t-test? Why are these assumptions necessary? How would violations to these assumptions affect the interpretation of the results of a t-test?

• Normality and independence are necessary for accurate p-values. Violations can lead to incorrect conclusions.

Explain the relationship between Type I errors, Type II errors, confidence level, and power. How do you calculate the probability of occurrence for each of these elements in an experiment?

• Type I Error (α): Rejecting a true null hypothesis.

• Type II Error (β): Failing to reject a false null hypothesis.

• Confidence Level: 1 - α.

• Power: 1 - β, or the ability to detect an effect.

If you did not adjust the critical values for a t-test (that is, you used the same critical values as a z-test – 1.64 or 1.96), would this affect the likelihood of committing a Type I or Type II Error? Explain why or why not.

• Yes, it would increase the likelihood of Type I errors because t-distribution critical values for small samples are higher.

What is the purpose of a power analysis? Explain the ways in which power can be increased or improved.

• Power analysis determines the required sample size to detect an effect.

• Power can be improved by increasing sample size, effect size, or alpha level.

How would small sample sizes affect a t-test and its assumptions? What about outliers?

• Small samples increase variability and reduce efficiency.

• Outliers can disproportionately affect results in small samples.