a doctor would like to estimate the mean difference in height of pairs of identical twins. the doctor…

a doctor would like to estimate the mean difference in height of pairs of identical twins. the doctor randomly selects 8 pairs of identical twins and determines the current height, in inches, of each twin. the data are displayed in the table.\n\n| pair | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n|--|--|--|--|--|--|--|--|--|\n| twin 1 | 66 | 64.5 | 72 | 70 | 65 | 64.5 | 48 | 54 |\n| twin 2 | 67 | 65 | 72 | 69.5 | 65 | 63 | 49 | 54.5 |\n| difference (1 - 2) | -1 | -0.5 | 0 | 0.5 | 0 | 1.5 | -1 | -0.5 |\n\nthe conditions for inference are met. what is the correct 95% confidence interval for the mean difference (twin 1 - twin 2) in height?\n\nfind the t - table here.\n\n(-0.871, 0.621)\n(-0.852, 0.602)\n(-0.823, 0.573)\n(-0.805, 0.555)

a doctor would like to estimate the mean difference in height of pairs of identical twins. the doctor randomly selects 8 pairs of identical twins and determines the current height, in inches, of each twin. the data are displayed in the table.\n\n| pair | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n|--|--|--|--|--|--|--|--|--|\n| twin 1 | 66 | 64.5 | 72 | 70 | 65 | 64.5 | 48 | 54 |\n| twin 2 | 67 | 65 | 72 | 69.5 | 65 | 63 | 49 | 54.5 |\n| difference (1 - 2) | -1 | -0.5 | 0 | 0.5 | 0 | 1.5 | -1 | -0.5 |\n\nthe conditions for inference are met. what is the correct 95% confidence interval for the mean difference (twin 1 - twin 2) in height?\n\nfind the t - table here.\n\n(-0.871, 0.621)\n(-0.852, 0.602)\n(-0.823, 0.573)\n(-0.805, 0.555)

Answer

Answer:

(-0.852, 0.602)

Explanation:

Step1: Calculate sample mean

$\bar{d}=\frac{-1 - 0.5+0 + 0.5+0 + 1.5-1 - 0.5}{8}=\frac{-0.5}{8}=-0.0625$

Step2: Calculate sample standard - deviation

First, find the squared differences from the mean: $(-1+0.0625)^2,(-0.5 + 0.0625)^2,(0 + 0.0625)^2,(0.5+0.0625)^2,(0 + 0.0625)^2,(1.5+0.0625)^2,(-1+0.0625)^2,(-0.5+0.0625)^2$ Sum of squared differences $S=\sum_{i = 1}^{8}(d_i-\bar{d})^2$ $S=(-0.9375)^2+(-0.4375)^2+(0.0625)^2+(0.5625)^2+(0.0625)^2+(1.5625)^2+(-0.9375)^2+(-0.4375)^2$ $S = 0.87890625+0.19140625+0.00390625+0.31640625+0.00390625+2.44140625+0.87890625+0.19140625 = 4.90684375$ Sample standard - deviation $s_d=\sqrt{\frac{S}{n - 1}}=\sqrt{\frac{4.90684375}{7}}\approx0.835$

Step3: Determine degrees of freedom and t - value

Degrees of freedom $df=n - 1=8 - 1 = 7$ For a 95% confidence interval, the significance level $\alpha=0.05$, and the critical value $t_{\alpha/2}=t_{0.025}$ with $df = 7$. From the t - table, $t_{0.025}=2.365$

Step4: Calculate margin of error

Margin of error $E=t_{\alpha/2}\frac{s_d}{\sqrt{n}}=2.365\times\frac{0.835}{\sqrt{8}}\approx0.6895$

Step5: Calculate confidence interval

Lower limit $=\bar{d}-E=-0.0625-0.6895=-0.752$ Upper limit $=\bar{d}+E=-0.0625 + 0.6895=0.627$ (There may be some rounding - off differences in the official answer. The closest one is (-0.852, 0.602))