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Explain the concept of the Central Limit Theorem (CLT) in the context of confidence intervals.
The CLT states that for a sufficiently large sample size (n โฅ 30), the sampling distribution of the sample means will be approximately normal, regardless of the shape of the population distribution. This allows us to use the t-distribution for constructing confidence intervals.
Explain the importance of random sampling when constructing a confidence interval.
Random sampling ensures that the sample is representative of the population, reducing bias and allowing for valid inferences about the population mean.
Explain why the t-distribution is used instead of the z-distribution when the population standard deviation is unknown.
The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample standard deviation. It has heavier tails, reflecting this increased uncertainty.
Explain the concept of independence in the context of confidence intervals.
Independence means that each observation in the sample does not influence any other observation. When sampling without replacement, the 10% condition (population size is at least 10 times the sample size) is used to ensure approximate independence.
Explain what a 95% confidence level means.
If we were to take many samples and construct a 95% confidence interval from each sample, approximately 95% of those intervals would contain the true population mean.
What is the t-distribution?
A probability distribution used for estimating population means when the sample size is small and the population variance is unknown. It has heavier tails than the normal distribution.
What are degrees of freedom (df)?
The number of independent pieces of information available to estimate a parameter. For a one-sample t-test, df = n - 1, where n is the sample size.
What is a point estimate?
A single value estimate for a population parameter. For estimating a population mean, the sample mean (xฬ) is used as the point estimate.
What is the margin of error?
The range of values above and below the sample statistic in a confidence interval. It quantifies the uncertainty in estimating the population parameter.
What is a confidence interval?
A range of values calculated from sample data that is likely to contain the true population parameter with a certain level of confidence.
What is the formula for a one-sample t-interval?
$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $t^*$ is the critical t-value, s is the sample standard deviation, and n is the sample size.
How do you calculate degrees of freedom for a one-sample t-test?
df = n - 1, where n is the sample size.
What is the formula for Margin of Error?
$Margin of Error = t^* \times \frac{s}{\sqrt{n}}$
What is the general structure of a confidence interval?
Point Estimate $\pm$ Margin of Error
How do you calculate the standard error when constructing a t-interval?
$SE = \frac{s}{\sqrt{n}}$, where s is the sample standard deviation and n is the sample size.