Revise later
SpaceTo flip
If confident
All Flashcards
What are the differences between population standard deviation (σ) and sample standard deviation (s) in the context of sampling distributions?
Population SD (σ): True variability in the entire population. | Sample SD (s): Estimate of population variability based on the sample.
What are the differences between using the t-distribution versus the z-distribution when analyzing the difference in two means?
t-distribution: Used when population standard deviations are unknown and estimated by sample standard deviations, especially with small sample sizes. | z-distribution: Used when population standard deviations are known or when sample sizes are very large.
What are the differences between a sampling distribution of a single mean and a sampling distribution of the difference between two means?
Single Mean: Distribution of sample means from one population. | Difference of Two Means: Distribution of the differences between sample means from two populations.
What are the differences between independent samples and dependent samples when comparing two means?
Independent Samples: Data from one sample does not influence the other. | Dependent Samples: Data from one sample is related to the other (e.g., matched pairs).
What are the differences between a parameter and a statistic in the context of comparing two populations?
Parameter: A numerical value that describes a characteristic of the entire population (e.g., the true difference in population means). | Statistic: A numerical value that describes a characteristic of a sample (e.g., the difference in sample means).
Explain the concept of the sampling distribution of the difference between two means.
It represents the distribution of differences in sample means (x̄1 - x̄2) from repeated samples. It allows us to make inferences about the true difference in population means (μ1 - μ2).
Explain the importance of the Central Limit Theorem (CLT) in the context of the difference between two means.
The CLT allows us to assume the sampling distribution of the difference in sample means is approximately normal when sample sizes are large (n ≥ 30 for both samples), even if the populations are not normally distributed.
Explain why variances are added when calculating the standard deviation of the difference between two means.
Variances are added because they represent the squared deviations from the mean, and adding them combines the variability from both distributions. Taking the square root then returns the combined standard deviation.
Explain the impact of sample size on the standard deviation of the sampling distribution.
Larger sample sizes decrease the standard deviation of the sampling distribution, leading to more precise estimates of the true difference in population means.
Explain the importance of checking conditions (normality or large sample size) before performing inference on two means.
Checking conditions ensures that the sampling distribution is approximately normal, which is a requirement for using normal-based statistical tests. Failing to check can lead to invalid conclusions.
What is the formula for the mean of the sampling distribution of the difference between two means?
μ(x̄1 - x̄2) = μ1 - μ2
What is the formula for the standard deviation of the sampling distribution of the difference between two means?
$$\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$
How do you calculate the standard deviation of the sampling distribution when population standard deviations are unknown?
Estimate using sample standard deviations: $$\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$
What is the formula to calculate the Z-score for the difference in sample means?
$$Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$.
What is the formula to calculate the degrees of freedom for the t-distribution when comparing two independent means?
The degrees of freedom can be approximated using Welch's t-test formula, which is complex and often calculated by statistical software. A conservative approach is to use the smaller of n1-1 and n2-1.