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What is the formula for the standard error of the slope?
\( SE(b) = \frac{s}{\sqrt{\sum(x_i - \bar{x})^2}} \), where s is the standard deviation of the residuals.
How do you calculate the margin of error for a confidence interval for the slope?
Margin of Error = t-score * Standard Error of the slope.
What is the general structure of a confidence interval?
Point Estimate ± Margin of Error
What is the formula to calculate degrees of freedom in linear regression?
df = n - 2, where n is the sample size.
How do you calculate the confidence interval for the slope of a regression line?
Sample Slope ± (t-critical value * Standard Error of the slope)
What are the differences between the t-distribution and the normal distribution?
T-distribution: Used when population standard deviation is unknown, heavier tails | Normal distribution: Used when population standard deviation is known, lighter tails.
What are the differences between a point estimate and a confidence interval?
Point estimate: Single value estimate of a parameter | Confidence interval: Range of plausible values for a parameter.
What are the differences between the standard deviation and the standard error of the slope?
Standard deviation: Measures the spread of data around the mean | Standard error of the slope: Measures the variability of the sample slope.
What are the differences between a 90% confidence interval and a 99% confidence interval?
90% CI: Narrower, less confidence | 99% CI: Wider, more confidence.
What are the differences between interpreting a confidence interval and interpreting a confidence level?
Confidence Interval: Provides a range for the true parameter. | Confidence Level: Indicates the success rate of the method if repeated many times.
Explain the concept of confidence level in the context of confidence intervals.
The percentage of times that the confidence interval will contain the true population parameter if the study is repeated many times.
Explain the importance of checking conditions before constructing a confidence interval.
Ensures the t-interval is valid and the results are reliable. Violating conditions can lead to inaccurate conclusions.
Explain how sample size affects the width of a confidence interval.
Larger sample sizes generally lead to narrower (more precise) confidence intervals.
Explain how the standard deviation of residuals affects the width of a confidence interval.
A larger standard deviation of residuals leads to a wider (less precise) confidence interval.
Explain the relationship between confidence level and the width of a confidence interval.
Higher confidence levels lead to wider confidence intervals because you need a larger range to be more confident you've captured the true population parameter.