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Why might random variation lead to errors in statistical inference?
Random variation can result in errors because it introduces variability in sample data, leading to estimates or decisions that may not accurately reflect the population.
How does the t-distribution differ from the normal distribution?
What happens to the t-distribution as the degrees of freedom increase?
What happens to the t-distribution as the degrees of freedom increase?
As the degrees of freedom increase, the t-distribution becomes closer in shape to the normal distribution, with less area in the tails.
Why is the t-distribution often used for estimating a population mean?
The t-distribution is used because the population standard deviation is typically unknown, making a one-sample t-interval appropriate for estimating the mean of a quantitative variable.
How do you find the critical value t* for a t-distribution?
The critical value t* can be found using a table of t-distribution values or by computer-generated output based on degrees of freedom.
Does a confidence interval for a population mean always contain the population mean?
No, a confidence interval either contains the population mean or it does not, as it is based on a random sample that varies.
What does it mean when we say we are "C% confident" about a confidence interval for a population mean?
It means we are C% confident that the interval captures the true population mean, acknowledging that a certain percentage of intervals will not contain the mean.
What should an interpretation of a confidence interval for a population mean include?
It should reference the sample taken and the population it represents, providing context for the interval estimate.
How can a confidence interval provide evidence to support a claim?
The interval gives a range of plausible values for the population mean, which may align with or refute the claim being investigated.
How does sample size affect the width of a confidence interval for a population mean?
As sample size increases, the width of the confidence interval tends to decrease, making the estimate more precise.
How is the width of a confidence interval related to sample size and confidence level?
The width of the interval is proportional to 1/√n and increases as the confidence level increases.
What is the appropriate test for a population mean with an unknown population standard deviation?
The appropriate test is a one-sample t-test for a population mean.
How can matched pairs be treated in significance testing?
Matched pairs can be treated as one sample of differences, and inference is conducted similarly to a one-sample t-test for a mean.
What is the null hypothesis for a one-sample t-test for a population mean?
The null hypothesis is H0: μ = μ0, where μ0 is the hypothesized population mean.
Why is the order of subtraction important when finding the mean difference in matched pairs?
The order affects the sign and interpretation of the difference, making it essential to define it consistently.
How should the p-value of a significance test for a population mean be interpreted?
The p-value is computed assuming the null hypothesis is true, indicating the probability of observing a result as extreme as the sample if H0 is correct.
What does a formal decision in a significance test compare?
It compares the p-value to the significance level (α). If p ≤ α, reject H0; if p > α, fail to reject H0.
How can the results of a significance test be used in research?
The results provide statistical reasoning to support or refute an answer to a research question about the sampled population.
What is the appropriate confidence interval procedure for comparing two independent samples?
A two-sample t-interval for the difference between population means is appropriate.
How is the margin of error for the difference of two sample means calculated?
It is calculated as the critical value (t*) times the standard error (SE) of the difference between the two means.
What is the point estimate for the difference of two population means?
The point estimate is the difference between the sample means, x̄1 − x̄2.
In repeated sampling, what proportion of confidence intervals will capture the true difference of population means?
Approximately C% of confidence intervals will capture the true difference, where C% is the confidence level.
What should an interpretation of a confidence interval for the difference between two population means include?
It should reference the samples taken and the populations they represent.
How can a confidence interval for the difference of population means support a claim?
The interval provides a range of plausible values that may align with or refute a contextual claim.
How does sample size affect the width of the confidence interval for the difference of two means?
As sample sizes increase, the width of the confidence interval decreases.
What is the appropriate test for a difference of two population means for a quantitative variable?
The appropriate test is a two-sample t-test for the difference between two population means.
What is the null hypothesis for a two-sample t-test for the difference between two population means?
The null hypothesis is H0: μ1 − μ2 = 0, or equivalently, H0: μ1 = μ2.
How should the p-value of a two-sample t-test for a difference between means be interpreted?
The p-value is computed assuming the null hypothesis is true, indicating the likelihood of observing the data if the population means are equal.
What formal decision is made in a two-sample t-test for a difference of means?
Compare the p-value to the significance level (α). If p ≤ α, reject H0; if p > α, fail to reject H0.
How can the results of a two-sample significance test support research conclusions?
The results provide evidence to answer research questions about the differences between the populations sampled.
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