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Confidence Intervals for Population Mean quiz

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  • What is the formula for the margin of error when constructing a confidence interval for a population mean with a known population standard deviation?
    The margin of error is calculated as e = zα/2 × σ/√n, where zα/2 is the critical z value, σ is the population standard deviation, and n is the sample size.
  • When should you use the t distribution instead of the z distribution for confidence intervals?
    Use the t distribution when the population standard deviation is unknown and you substitute the sample standard deviation (s) in the calculation.
  • What are the two main conditions that must be checked before constructing a confidence interval for a population mean?
    The sample must be random, and either the population is normal or the sample size n is greater than 30.
  • How do you interpret a 90% confidence interval for a population mean?
    You can say you are 90% confident that the true population mean lies within the calculated interval.
  • What is the formula for the margin of error when using the t distribution?
    The margin of error is e = tα/2 × s/√n, where tα/2 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 the t distribution in confidence intervals?
    Degrees of freedom are calculated as n - 1, where n is the sample size.
  • How does the t distribution compare to the normal distribution as sample size increases?
    As sample size increases, the t distribution becomes more similar to the normal distribution.
  • What is the process for finding the critical t value using a calculator?
    Use the inverse t function, inputting the area to the left (alpha/2) and the degrees of freedom (n-1), to find the critical t value.
  • What is the effect on the confidence interval when using the t distribution instead of the z distribution?
    The confidence interval is wider with the t distribution because the t critical value is larger, reflecting more uncertainty.
  • What steps do you follow to construct a confidence interval for the mean when given statistics on a TI-84 calculator?
    Go to the tests menu, choose z interval if sigma is known or t interval if not, enter the statistics, set the confidence level, and calculate.
  • How do you construct a confidence interval for the mean when given raw data on a TI-84 calculator?
    Enter the data into a list, go to the tests menu, select t interval, choose data, set the confidence level, and calculate.
  • What is the formula for alpha when calculating a confidence interval?
    Alpha is calculated as 1 minus the confidence level (α = 1 - C).
  • How do you find the lower and upper bounds of a confidence interval?
    Subtract the margin of error from the sample mean for the lower bound and add it for the upper bound.
  • Why is it important to interpret a confidence interval after constructing it?
    Interpreting the interval allows you to communicate the level of confidence that the population mean lies within the calculated range.
  • What is the main difference in calculation between a z interval and a t interval for confidence intervals?
    A z interval uses the population standard deviation and critical z value, while a t interval uses the sample standard deviation and critical t value.