Chapter 10 Estimating with Confidence Vocabulary: Confidence interval significance level Margin of error Interval Confidence level Critical value A level C confidence interval test statistic z test statistic t distribution z distribution paired t procedures Test of significance Upper p critical value standard error Z* margin of error degrees of freedom Robust Calculator Skills: Z interval Z-test T interval 10-1 Confidence Intervals: The Basics (617-642) 1. In statistics, what is meant by a 95% confidence interval? 95% of samples will capture the true mean, parameter 2. Sketch and label a 95% confidence interval for the standard normal curve. 3. In a sampling distribution of x , why is the interval of numbers between x 2s called a 95% confidence interval? 95% of Normal population is within 2 standard deviations of the mean 4. Define a level C confidence interval. Estimate ± margin of error 5. Sketch and label a 90% confidence interval for the standard normal curve. 6. What does z* represent? The critical value of z or t. ( # of standard deviations) 7. What is the value of z* for a 90% confidence interval? Include a sketch. 1.645 8. What is the value of z* for a 95% confidence interval? Include a sketch. 1.96 9. What is the value of z* for a 99% confidence interval? Include a sketch. 2.576 10. What is meant by the upper p critical value of the standard normal distribution? Area to the right of a positive z* value 11. Explain how to find a level C confidence interval for an SRS of size n having unknown mean and known standard * deviation . The formula to find the confidence interval is X z n 12. What is meant by a margin of error? It means how far “off” you are from the true mean, parameter. 13. Why is it best to have high confidence and a small margin of error? It will give a closer approximation to the true mean. 14. What happens to the margin of error as z* decreases? Does this result in a higher or lower confidence level? The margin of error is the formula: * z As z* decreases, the margin of error decreases and lowers confidence. n 15. What happens to the margin of error as decreases? As σ decreases, the margin of error decreases and lowers confidence. 16. What happens to the margin of error as n increases? By how many times must the sample size increase in order to cut the margin of error in half? As n increases, the margin of error decreases. The sample size must increase by 4 times to decrease the margin of error in half. 17. The formula used to determine the sample size that will yield a confidence interval for a population mean with a specified margin of error m is z * m . Solve for n. n z* m n m n z * z* m n z * n m 2 z * n m 10-2 Estimating a Population Mean (642 - 662) 1. Under what assumptions is s a reasonable estimate of σ? 1. If it is an SRS. 2. If the samples are independent. N ≥10n 3. If the sampling distribution is normal. np ≥ 10 and n(1-p) ≥ 10 2. In general, what is meant by the standard error of a statistic? It is the standard deviation of a statistic. 3. What is the standard deviation of the sample mean x ? It is . n s 4. What is the standard error of the sample mean x ? It is . n 5. Describe the similarities between a standard normal distribution and a t distribution? The distributions have similar shape. They are symmetric, bell-shaped and unimodal. 6. Describe the differences between a standard normal distribution and a t distribution? The spread of the t-distribution is larger. 7. How do you calculate the degrees of freedom for a t distribution? Degrees of Freedom is found by subtracting one from the sample size. The formula is: df = n – 1. 8. What happens to the t distribution as the degrees of freedom increase? As n increases, the degrees of freedom increases and the t distribution becomes more normal. 9. How would you construct a level C confidence interval for μ if σ is unknown? * s X t n 10. The z-table gives the area under the standard normal curve to the left of z. What does the t-table give? It gives the area to the right. 11. In a matched pairs t procedure, what is μ, the parameter of interest? The parameter of interest is the mean difference in responses of two treatments. For example: difference after before 12. Samples from normal distributions have very few outliers. If your data contains outliers, what does this suggest? It suggests that the parent population is non-normal. 13. If the size of the SRS is less than 15, when can we use t procedures on the data? If the data are clearly normal or no obvious outliers. 14. If the size of the SRS is at least 15, when can we use t procedures on the data? We can use t procedures unless there are outliers or strong skewness. 15. If the size of the SRS is at least 30 or 40, when can we use t procedures on the data? If the data is at least 30 or 40 then t distribution can be used. 10-3 Estimating a Population Proportion (663- 677) 1. In statistics, what is meant by a sample proportion? The statistic that estimates the parameter, p, of the sample proportion. 2. Give the mean and standard deviation for the sampling distribution of pˆ ? p(1 p ) The mean is p. p . The standard deviation is . p p n 3. How do you calculate the standard error of pˆ ? p(1 p ) The standard error of p , SE is: SE p p n 4. What assumptions must be met in order to use z procedures for inference about a proportion? 1. If it is an SRS. 2. If the samples are independent. N ≥10n 3. If the sampling distribution is normal. np ≥ 10 and n(1-p) ≥ 10 5. Describe how to construct a level C confidence interval for a population proportion. p(1 p ) p z* n 6. What formula is used to determine the sample size necessary for a given margin of error? p*(1 p *) The margin of error formula is m z * . Solving this for n, the sample size, is n (z *)2 p * (1 p *) n . Where p* is an estimated value of the sample proportion. m2.