All Flashcards
Explain the concept of the impact of sample size on the margin of error.
As sample size increases, both the critical value (t*) and the standard error decrease, leading to a narrower confidence interval and a more precise estimate of the population mean.
Explain the concept of using a t-distribution instead of a z-distribution.
Use a t-distribution when the population standard deviation is unknown and you are estimating it with the sample standard deviation. Use a z-distribution when the population standard deviation is known.
Explain how to test a claim about a population mean using a confidence interval.
If the claimed population mean falls within the confidence interval, you cannot reject the claim. If it falls outside, you have reason to doubt the claim.
Explain the importance of random sampling when constructing a confidence interval.
Random sampling ensures that the sample is representative of the population, which is a key assumption for the validity of the confidence interval. It reduces bias and allows for accurate inferences about the population mean.
Explain the concept of degrees of freedom.
Degrees of freedom (df) represent the number of independent pieces of information available to estimate a parameter. In the context of a t-distribution, df = n - 1, where n is the sample size.
What are the differences between t-distribution and z-distribution?
t-distribution: Used when population standard deviation is unknown, has heavier tails, depends on degrees of freedom. | z-distribution: Used when population standard deviation is known, standard normal distribution.
What are the differences between a point estimate and a confidence interval?
Point estimate: A single value estimate of a population parameter. | Confidence interval: A range of values likely to contain the population parameter.
What are the differences between standard deviation and standard error?
Standard deviation: Measures the spread of data in a sample. | Standard error: Measures the accuracy of the sample mean as an estimate of the population mean.
What are the differences between increasing sample size and increasing confidence level?
Increasing sample size: Decreases the width of the confidence interval. | Increasing confidence level: Increases the width of the confidence interval.
What are the differences between Type I and Type II error?
Type I error: Rejecting a true null hypothesis (false positive). | Type II error: Failing to reject a false null hypothesis (false negative).
What is the formula for standard error (SE)?
where s is the sample standard deviation and n is the sample size.
What is the general formula for a confidence interval?
Point estimate ± (critical value)(standard error)
Formula for degrees of freedom (df) when estimating population mean?
df = n - 1, where n is the sample size.
How do you calculate the margin of error for a confidence interval?
Margin of Error = Critical Value (t*) * Standard Error
What is the formula for the t-score?