Chapter 17, Problem 17.16AYK

To determine

To find:that is there is proof of a bionic pancreas that supports patients reach a lower level of plasma glucose than an insulin pump.

Answer to Problem 17.16AYK

Glucose level in insulin pump is greater than in bionic pancreas.

Explanation of Solution

Given:

Subject	Control	Bionic
1	220	136
2	202	142
3	188	149
4	189	147
5	179	140
6	172	134
7	164	132
8	161	139
9	159	154
10	95	136
11	119	152
12	122	117
13	133	151
14	138	119
15	144	156
16	157	151
17	222	138
18	181	137
19	169	142
20	163	125
21	156	144
22	153	136
23	152	156
24	129	136
25	138	179
26	147	139
27	147	135
28	141	150
29	159	134
30	158	146
31	151	144
32	145	139

Formula used:

  $\overline{x}=\frac{{\displaystyle \sum d}}{n}$

  $s=\sqrt{\frac{{{\displaystyle \sum \left(d-\overline{x}\right)}}^2}{n-1}}$

  $t=\frac{\overline{d}}{\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$\sqrt{n-1}$}\right.}$

Calculation:

Subject	Control	Bionic	d=Control-Bionic	$d-\overline{d}$	${\left(d-\overline{d}\right)}^2$
1	220	136	84	67.81	4598.54
2	202	142	60	43.81	1919.54
3	188	149	39	22.81	520.41
4	189	147	42	25.81	666.29
5	179	140	39	22.81	520.41
6	172	134	38	21.81	475.79
7	164	132	32	15.81	250.04
8	161	139	22	5.81	33.79
9	159	154	5	-11.19	125.16
10	95	136	-41	-57.19	3270.41
11	119	152	-33	-49.19	2419.41
12	122	117	5	-11.19	125.16
13	133	151	-18	-34.19	1168.79
14	138	119	19	2.81	7.91
15	144	156	-12	-28.19	794.54
16	157	151	6	-10.19	103.79
17	222	138	84	67.81	4598.54
18	181	137	44	27.81	773.54
19	169	142	27	10.81	116.91
20	163	125	38	21.81	475.79
21	156	144	12	-4.19	17.54
22	153	136	17	0.81	0.66
23	152	156	-4	-20.19	407.54
24	129	136	-7	-23.19	537.66
25	138	179	-41	-57.19	3270.41
26	147	139	8	-8.19	67.04
27	147	135	12	-4.19	17.54
28	141	150	-9	-25.19	634.41
29	159	134	25	8.81	77.66
30	158	146	12	-4.19	17.54
31	151	144	7	-9.19	84.41
32	145	139	6	-10.19	103.79
	Sum		518		28200.88

  $\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum d}}{n}=\frac{518}{32}=16.19\\ s=\sqrt{\frac{{{\displaystyle \sum \left(d-\overline{x}\right)}}^2}{n-1}}=\sqrt{\frac{28200.88}{32-1}}=\text{30}\text{.1613}\end{array}$

Null Hypotheses: - glucose level is the equal in both from bionic pancreas and in insulin pump.

  $H_0:{\mu }_1={\mu }_2$

Alternative Hypotheses: - glucose level in insulin pump is greater than in bionic pancreas.

  $H_1:{\mu }_1>{\mu }_2$ (one-tailed)

Level of significance is 0.05

Test statistics

  $t=\frac{\overline{d}}{\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$\sqrt{n-1}$}\right.}=\frac{16.19}{\raisebox{1ex}{$\text{30}\text{.1613}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{31}$}\right.}=\text{2}\text{.99}$

  ${\text{t}}_{0.05,df=31}=1.696$

Conclusion Calculated test statistics is greater than critical value, therefore reject the null hypotheses, hence conclude that glucose level in insulin pump is greater than in bionic pancreas.