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Confidence Intervals

Chapter 6. Elementary Statistics Larson Farber. Confidence Intervals. Inferential Statistics. Estimating Parameters Hypothesis Testing. Inferential Statistics-the branch of statistics that uses sample statistics to make inferences about population parameters.

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Confidence Intervals

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  1. Chapter 6 Elementary Statistics Larson Farber Confidence Intervals

  2. Inferential Statistics Estimating Parameters Hypothesis Testing Inferential Statistics-the branch of statistics that uses sample statistics to make inferences about population parameters. Applications of Inferential Statistics

  3. Point Estimate DEFINITION: A point estimate is a single value estimate for a population parameter. The best point estimate of the population mean  is thesample mean .

  4. Example: Point Estimate • A random sample of airfare prices (in dollars) for a one-way ticket from Atlanta to Chicago. Find a point estimate for the population mean, . 99 102 105 105 104 95 100 114 108 103 94 105 101 109 103 98 96 98 104 87 101 106 103 90 107 98 101 107 105 94 111 104 87 117 101 The sample mean is The point estimate for the price of all one way tickets from Atlanta to Chicago is $101.77.

  5. • • 101.77 101.77 ) ( Interval Estimates Point estimate An interval estimate is an interval, or range of values used to estimate a population parameter The level of confidence, c is the probability that the interval estimate contains the population parameter

  6. Distribution of Sample Means When the sample size is at least 30, the sampling distribution for is normal Sampling distribution 0.025 0.025 z -1.96 0 1.96 For c = 0.95 0.95 95% of all sample means will have standard scores between z = -1.96 and z = 1. 96

  7. DEFINITION: Given a level of confidence, c, the maximum error of estimate E is the greatest possible distance between the point estimate and the value of the parameter it is estimating. When n  30, the sample standard deviation, s can be used for . Maximum Error of Estimate • Find E, the maximum error of estimate for the one-way plane fare from Atlanta to Chicago for a 95% level of confidence given s = 6.69 Using zc=1.96, s = 6.69 and n = 35, You are 95% confident that the maximum error of estimate is $2.22

  8. Confidence Intervals for Definition: A c-confidence interval for the population mean is • 101.77 You found = 101.77 and E = 2.22 Right endpoint Left endpoint 103.99 99.55 ( ) • Find the 95% confidence interval for the one-way plane fare from Atlanta to Chicago. With 95% confidence, you can say the mean one-way fare from Atlanta to Chicago is between$99.55and$103.99

  9. Sample Size Given a c-confidence level and an maximum error of estimate, E, the minimum sample size n, needed to estimate , the population mean is • You want to estimate the mean one-way fare from Atlanta to Chicago. How many fares must be included in your sample if you want to be 95% confident that the sample mean is within $2 of the population mean? You should include at least 43 fares in your sample. Since you already have 35, you need 8 more.

  10. The t-distribution Sampling distribution n =13 .90 d.f.=12 c=90% .05 .05 t -1.782 0 1.782 If the distribution of a random variable x is normal and n < 30, then the sampling distribution of is a t-distribution with n-1 degrees of freedom. The critical value for t is 1.782. 90% of the sample means with n = 12 will lie between t = -1.782 and t = 1.782

  11. Confidence Interval-Small Sample Maximum error of estimate 1. The point estimate is x = 4.3 pounds • In a random sample of 13 American adults, the mean waste recycled per person per day was 4.3 pounds and the standard deviation was 0.3 pound. Assume the variable is normally distributed and construct a 90% confidence interval for . 2. The maximum error of estimate is

  12. Confidence Interval-Small Sample 1. The point estimate is x=4.3 pounds 2. The maximum error of estimate is Left endpoint Right endpoint • ( ) 4.152 4.3 4.448 4.15 <  < 4.45 With 90% confidence, you can say the mean waste recycled per person per day is between4.15and4.45 pounds.

  13. Confidence Intervals for Population Proportions The point estimate for p, the population proportion of successes is given by the proportion of successes in a sample (Read as p-hat) is the point estimate for the proportion of failures where If np  5 and nq  5 , the sampling distribution for is normal.

  14. Confidence Intervals for Population Proportions The maximum error of estimate, E for a c-confidence interval is: A c-confidence interval for the population proportion, p is

  15. Confidence Interval for p 1. The point estimate for p is 3. • In a study of 1907 fatal traffic accidents, 449 were alcohol related. Construct a 99% confidence interval for the proportion of fatal traffic accidents that are alcohol related. 2. 1907(.235) 5 and 1907(.765) 5, so the sampling distribution is normal.

  16. Confidence Interval for p Right endpoint Left endpoint ) ( .21 • .26 .235 • In a study of 1907 fatal traffic accidents, 449 were alcohol related. Construct a 99% confidence interval for the proportion of fatal traffic accidents that are alcohol related. 0.21 < p < 0.26 With 99% confidence, you can say the proportion of fatal accidents that are alcohol related is between 21% and 26%.

  17. Minimum Sample Size If you do not have a preliminary estimate, use 0.5 for both If you have a preliminary estimate for p and q the minimum sample size given a c-confidence interval and a maximum error of estimate needed to estimate p is:

  18. Example: Minimum Sample Size With no preliminary estimate use 0.5 for • You wish to estimate the proportion of fatal accidents that are alcohol related at a 99% level of confidence. Find the minimum sample size needed to be be accurate to within 2% of the population proportion. = You will need at least 4415 for your sample.

  19. Example: Minimum Sample Size • You wish to estimate the proportion of fatal accidents that are alcohol related at a 99% level of confidence. Find the minimum sample size needed to be be accurate to within 2% of the population proportion. Use a preliminary estimate of p = 0.235 With a preliminary sample you need at least n =2981for your sample.

  20. The Chi-Square Distribution The point estimate for 2 is s2 and the point estimate for  is s. If the sample size is n, use a chi-square 2 distribution with n-1 d.f. to form a c-confidence interval. .95 6.908 28.845 • Find R2 the right- tail critical value and L2 the left-tail critical value for c = 95% and n = 17. When the sample size is 17, there are 16 d.f. Area to the right of R2 is (1- 0.95)/2 = 0.025 and area to the right of L2 is (1+ 0.95)/2 = 0.975 L2=6.908 R2=28.845

  21. Confidence Intervals for • You randomly select and record the prices of 17 CD players. The sample standard deviation is $150. Construct a 95% confidence interval for You can say with 95% confidence that is between 12480.50 and 52113.49 and between $117.72 and $228.28. A c-confidence interval for a population variance is: To estimate the standard deviation take the square root of each endpoint. Find the square root of each endpoint

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