CHAPTER 6 Confidence Intervals 6 1 a y 1269 s 145 n 8 The standard error of the mean is 145 s 51 3 ng gm SE y n 8 b y 1269 s 145 n 30 The standard error of the mean is 145 s 26 5 ng gm SE y n 30 6 2 a 15 25 3 0 cm b 15 100 1 5 cm 6 3 y 9 520 s 1 429 SE 1 429 5 6391 64 gm kg 6 4 a 3 06 86 33 mm b 25 Frequency 20 15 10 5 52 56 60 64 68 Tail length mm SE s 6 5 a We would predict the SD of the new measurements to be about 3 mm because this is our estimate based on Exercise 6 4 of the population SD b We would expect the SE of the new measurements to be 3 500 13 mm 6 6 To convey the homogeneity of the group of rats the 10 gm should be the SD since the SD describes variability among the rats The SE describes the precision of the sample mean but this depends on the sample size 85 6 7 a the SE b the SD c the SE 6 8 and 6 9 See Section III of this Manual 6 10 a y 31 720 mg s 8 729 mg n 5 The standard error of the mean is 8 729 s 3 904 3 9 mg SE y n 5 b The degrees of freedom are n 1 5 1 4 The critical value is t 05 2 132 The 90 confidence interval for is y t s 05 n 8 729 31 720 2 132 5 23 4 40 0 or 23 4 40 0 mg 6 11 a The degrees of freedom are n 1 5 1 4 The critical value is t 025 2 776 The 95 confidence interval for is s y t 025 n 8 729 31 720 2 776 5 20 9 42 5 or 20 9 42 5 mg b We are 95 confident that the mean thymus gland weight in the population of chick embryos is between 20 9 and 42 5 mg 6 12 a y 28 7 s 4 5898 SE 4 5898 6 1 87 1 9 g ml 28 7 2 571 1 9 23 8 33 6 or 23 8 33 6 g ml b mean blood serum concentration of Gentamicin 1 5 hours after injection of 10 mg kg body weight in healthy three year old female Suffolk sheep c No The 95 refers to the percentage in a meta experiment of confidence intervals that would contain Since the width of a confidence interval depends on n the percentage of observations contained in the confidence interval also depends on n and would be very small if n were large 6 13 a This statement is false The confidence interval allows us to make an inference concerning the mean of the entire population We know that 59 77 y 61 09 b This statement is true See part a 6 14 This statement is false The confidence interval concerns the mean of the population It does not tell us where individual data points lie 6 15 The higher the confidence level the greater the length of the interval Thus 90 85 and 80 confidence intervals correspond to 822 858 824 856 and 826 854 respectively 86 6 16 a y 13 0 s 12 4 n 10 The degrees of freedom are n 1 10 1 9 The critical value is t 05 2 262 The 95 confidence interval for is s y t 025 n 12 4 13 0 2 262 10 4 1 21 9 or 4 1 21 9 pg ml b We are 95 confident that the average drop in HBE levels from January to May in the population of all participants in physical fitness programs like the one in the study is between 4 1 and 21 9 pg ml 6 17 A histogram and normal probability plot of the data shown here support the use of a normal curve model for these data 30 20 10 0 20 0 10 20 30 40 1 0 1 nscores 6 18 a 5 111 2 306 818 9 4482 5740 or 4 482 5 740 units b We are 95 confident that the average invertase activity of all fungal tissue incubated at 95 relative humidity for 24 hours is between 4 482 units and 5 740 units c To check that the data are from a normal population we can make a normal probability plot 6 19 6 21 2 042 1 84 36 5 58 6 84 or 5 58 6 84 g dl 6 20 y 1 20 s 14 n 50 The degrees of freedom are 50 1 49 From Table 4 with df 50 the df value closest to 49 we find that t 05 1 676 The 90 confidence interval for is s y t 05 n 14 1 20 1 676 50 1 17 1 23 or 1 17 1 23 mm 6 21 We are 95 confident that the mean Bayley Index of prematurely born infants who receive intravenousfeeding solutions is between 93 8 and 102 1 Although the center of the interval is 97 95 which is less 87 than the general population average of 100 the interval extends above 100 so we cannot be sure that is less than 100 6 22 y 10 3 s 0 9 n 101 There are 100 degrees of freedom so t 025 1 984 The 95 confidence interval for is 9 10 3 1 984 101 10 12 10 48 or 10 12 10 48 g dLi 6 23 1 025 975 In Table 3 z 1 96 corresponds to an area of 975 A t distribution with df is a normal distribution 6 24 1 0025 9975 In Table 3 an area of 9975 corresponds to z 2 81 A t distribution with df is a normal distribution thus t 0025 2 81 when df 6 25 The confidence coefficient is approximately 68 because a t distribution with df is a normal distribution and the area under a normal curve between z 1 00 and z 1 00 is approximately 68 6 26 a Smaller The area included between t 1 00 and t 1 00 is smaller for a Student s t distribution than for a normal distribution b It is not affected because of the Central Limit Theorem 6 27 a Guessed SD 20 kg n must satisfy the inequality 20 5 n so n 16 b n must satisfy the inequality 40 5 n so n 64 The required sample size does not double but rather is four times as large 6 28 We use the inequality Guessed SD Desired SE n In this case the desired SE is 3 mg dl and the guessed SD is 40 mg dl Thus the inequality is 40 40 3 or 3 n which means that n 177 8 so a sample of …
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