Chapter 10: Estimating with Confidence
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1 Key Vocabulary: confidence interval margin of error interval confidence level a level C confidence interval degrees of freedom standard error t distribution one-sample t statistic robust power upper p critical value p-value acceptance sampling statistically significant significance level Calculator Skills: ZInterval TInterval tpdf (X, df) normalpdf (X) 10.1 Confidence Intervals: The Basics (pp ) 1. In statistics, what is meant by a 95% confidence interval? 2. A confidence interval takes the form of : estimate ± margin of error where: estimate = margin of error = 3. Define a level C confidence interval. 4. Sketch and label a 95% confidence interval for the standard normal curve.
2 5. In a sampling distribution of x, why is the interval of numbers between x ± 2s called a 95% confidence interval? 6. Sketch and label a 90% confidence interval for the standard normal curve. 7. State the three conditions for constructing a confidence interval for µ. 8. What does z* represent? 9. What is the value of z* for a 95% confidence interval? Include a sketch. 10. What is the value of z* for a 90% confidence interval? Include a sketch. 11. What is the value of z* for a 99% confidence interval? Include a sketch. 12. What is meant by the upper p critical value of the standard normal distribution? 13. Explain how to find a level C confidence interval for an SRS of size n having unknown mean µ and known standard deviation σ.
3 14. Use of the Inference Toolbox is extremely important when using inference. Summarize the four steps of the inference toolbox. 15. Why is it best to have high confidence and a small margin of error? 16. What happens to the margin of error as z* gets smaller? Does this result in a higher or lower confidence level? 17. What happens to the margin of error as σ gets smaller? 18. What happens to the margin of error as n gets larger? By how many times must the sample size n increase in order to cut the margin of error in half? 19. The formula used to determine the sample size n that will yield a confidence interval for a population mean with a specified margin of error m is z* σ m. Solve for n. n
4 10.2 Estimating a Population Mean (pp ) 1. Under what conditions can we use s as a reasonable estimate of σ? 2. In general, what is meant by the standard error of a statistic? 3. How does the standard deviation differ to of the sample mean x? 4. What is the standard error of the sample mean x? 5. How do you calculate the degrees of freedom for a t distribution? 6. Describe the similarities between a standard normal distribution and a t distribution. 7. Describe the differences between a standard normal distribution and a t distribution. 8. What happens to the t distribution as the degrees of freedom increase? 9. How would you construct a level C confidence interval for µ if σ is unknown?
5 10. In a matched pairs t procedure, what is µ, the parameter of interest? 11. What does it mean if an inference procedure is robust? 12. If the size of the SRS is less than 15, when can we use t procedures on the data? 13. If the size of the SRS is at least 15, when can we use t procedures on the data? 14. If the size of the SRS is at least 30, when can we use t procedures on the data?
6 10.3 Estimating a Population Proportion (pp ) 1. In statistics, what is meant by a sample proportion: ˆp? 2. Give the mean and standard deviation for the sampling distribution of ˆp? 3. How does the standard deviation differ to to standard error for the sampling distribution of ˆp? 4. How do you calculate the standard error of ˆp? 5. What conditions must be met in order to use z procedures for inference about a proportion? 6. Describe how to construct a level C confidence interval for a population proportion. 7. What formula is used to determine the sample size necessary for a given margin of error?
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