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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 ≥ 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 are the differences between the t-distribution and the normal distribution?
t-distribution: Used when σ is unknown, heavier tails, shape depends on degrees of freedom. | Normal distribution: Used when σ is known, lighter tails, defined by mean and standard deviation.
What are the differences between a t-test and a z-test?
t-test: Used when population standard deviation is unknown, uses t-distribution, appropriate for small sample sizes. | z-test: Used when population standard deviation is known, uses normal distribution, appropriate for large sample sizes.
What are the differences between increasing the sample size and increasing the confidence level on the width of a confidence interval?
Increasing sample size: Decreases width of confidence interval, reduces margin of error. | Increasing confidence level: Increases width of confidence interval, increases margin of error.
What are the differences between sample mean and population mean?
Sample mean: Calculated from a subset of the population, used to estimate population mean, varies from sample to sample. | Population mean: True average of the entire population, usually unknown, constant value.
What are the differences between standard deviation and standard error?
Standard deviation: Measures the spread of individual data points around the sample mean. | Standard error: Measures the spread of sample means around the population mean, estimates the variability of the sample 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 (x̄) 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.