Section 9.1 Day 4 CI for a Mean
t* Why do we use a t-distribution to find a confidence interval (CI) for a mean when σ is unknown?
t* The t-distribution is more robust than a z-distribution in that the t-distribution compensates for the skewed sampling distribution of s.
To increase the width of the interval, replace z To increase the width of the interval, replace z* with a larger value called t*. x z* x t*
What is the 2nd condition we check to determine if it is appropriate to use a confidence interval to estimate a population mean with an unknown population standard deviation?
What is the 2nd condition we check to determine if it is appropriate to use a confidence interval to estimate a population mean with an unknown standard deviation? We check to see if it is reasonable to assume the sample data came from a normally distributed population.
Is the 2nd condition satisfied for this set of sample data? 3.7, 4.3, 4.9, 5.1, 5.4, 5.9, 6.1, 6.5, 7.0, 7.1, 7.5, 8.5, 8.8, 9.3, and 9.9
Use Modified Boxplot Is the 2nd condition satisfied for this set of sample data? 3.7, 4.3, 4.9, 5.1, 5.4, 5.9, 6.1, 6.5, 7.0, 7.1, 7.5, 8.5, 8.8, 9.3, and 9.9
Yes, because the sample data is fairly symmetric with no outlier it is reasonable to assume the population is normally distributed.
Yes, because the sample data is fairly symmetric with no outlier it is reasonable to assume the population is normally distributed.
Page 574, P7 CI from P4 is (4.229, 5.809).
Page 574, P7 D. You are 95% confident that the mean aldrin level of the Wolf River falls in the confidence interval (4.229, 5.809)
Page 574, P5
Page 574, P5 TInterval Inpt: Data Stats x: 98.1 sx: 0.73 n: 122 C-Level: .95 Calculate
Page 574, P5 TInterval Inpt: Data Stats x: 98.1 sx: 0.73 n: 122 C-Level: .95 Calculate (97.969, 98.231)
Page 574, P5 b. I’m 95% confident that the true mean body temperature of all men is between 97.969o F and 98.231o F.
Page 574, P5 c. 98.6o F is not in this interval. (97.969, 98.231)
Page 574, P5 c. 98.6o F is not in this interval. We can conclude the mean body temperature of men is less than 98.6o F. (97.969, 98.231)
Page 575, P9 Read problem. Then refer to page 567, parts a and b under Solution section.
Page 575, P9 males: (97.48, 98.28) E = ? females: (98.14, 98.90) E = ?
Page 575, P9 males: (97.48, 98.28) E = 0.4 females: (98.14, 98.90) E = ?
Page 575, P9 males: (97.48, 98.28) E = 0.4 females: (98.14, 98.90) E = 0.38
Page 575, P9 Margin of error is larger for males because the standard deviation of the sample of males is larger than that for the sample of females. x ± t*● smale = 0.555 sfemale = 0.527
Page 578, E9
Page 578, E9 Which CI is widest? a. 95% interval with n = 4 and s = 10; 90% interval with n = 5 and s = 9; 99% interval with n = 4 and s = 10
Page 578, E9
Page 578, E9 Which CI is widest? b. 95% interval with n = 3 and s = 10; 95% interval with n = 4 and s = 10; 95% interval with n = 5 and s = 10
Page 578, E9
Page 578, E9 Which CI is widest? c. 90% interval with n = 10 and s = 5; 95% interval with n = 10 and s = 5; 95% interval with n = 10 and s = 10
Page 578, E9
Page 579, E10
Page 579, E10 A. They would have identical values for the lower and upper limits of the confidence interval.
Page 579, E10 A. They would have identical values for the lower and upper limits of the confidence interval. False: CI = x ± t*● x and s both vary by sample
Page 579, E10 B. They would have the same margin of error.
Page 579, E10 B. They would have the same margin of error. False; E = t* ● s varies by sample
Page 579, E10 C. The confidence intervals would have the same center but different widths.
Page 579, E10 C. The confidence intervals would have the same center but different widths. False; center is x which varies by sample.
Page 579, E10 D. None of the above is true is the correct answer.
z* When would we use z* to construct a confidence interval?
z* When would we use z* to construct a confidence interval? If the sampling distribution has an approximately normal distribution (i.e. no evidence of skewness and no outliers) and we know σ.
Questions?
Page 574, P4
Page 574, P4 Interval from P1 is (4.335, 5.704)
Page 574, P4 Interval from P1 is (4.335, 5.704) Why is the second interval wider?
Page 574, P4 Interval from P1 is (4.335, 5.704) Why is the second interval wider? Because P1 used z* and P4 used t*