9.1: z-based confidence intervals for a population mean: SD known Flashcards
What is a confidence interval?
A confidence interval is a range of values, derived from the sample statistics, used to estimate the true population parameter.
It’s constructed so there’s a specified probability that the confidence interval contains the true parameter.
What does a 95% confidence interval imply?
A 95% confidence interval implies that if we were to take many samples and build a confidence interval from each, about 95% of these intervals would contain the true population mean.
What is a confidence level?
The confidence level is the percentage (e.g., 95%, 99%) that expresses the degree of certainty in the confidence interval estimation.
It is the probability that the interval contains the true population parameter.
How do you calculate the standard error of the mean, σx̄?
The standard error of the mean (σx̄) is calculated using the formula:
σx̄ = σ / √n,
where σ is the population standard deviation and n is the sample size.
What is the formula for a z-based confidence interval when σ is known?
What is the z-score for a 95% confidence interval?
The z-score for a 95% confidence interval is approximately ±1.96, which corresponds to the 95th percentile of the standard normal distribution.
How can confidence intervals be used in a practical context?
Confidence intervals can be used to make decisions or to make assertions about a population parameter with a known degree of certainty.
For example, an automaker can use a 95% confidence interval to determine whether a car model’s mean mileage meets federal requirements for a tax credit.
What is the general procedure for calculating a confidence interval for a population mean?
The general procedure is a three-step process:
Decide on the confidence coefficient and find the corresponding z-scores that capture the central area under the standard normal curve equal to the confidence coefficient.
Calculate the standard error of the mean using the population standard deviation and sample size.
Form the confidence interval using the sample mean and standard error with the selected z-scores.
What is the z-score for a 99% confidence interval?
The z-score for a 99% confidence interval is approximately ±2.575, which corresponds to the critical values that have 0.5% of the area in each tail of the standard normal distribution.
What is the formula for a 99% confidence interval when the population standard deviation (σ) is known?
What is the margin of error and how does it differ between the 95% and 99% confidence intervals?
The margin of error is the extent of the interval on either side of the sample mean and represents the maximum expected difference between the true population parameter and a sample estimate.
The margin of error for a 99% confidence interval is larger than for a 95% confidence interval, reflecting increased certainty.
Can the confidence interval formula still be used if the population is not normally distributed?
Yes, the confidence interval formula based on the standard normal distribution can still be used for large sample sizes (typically n ≥ 30) due to the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normal regardless of the population distribution.
What is the general formula for a confidence interval for a population mean when σ is known?
How do you adjust the confidence interval for different levels of confidence?
