Question

Construct a​ 90% confidence​ interval, using the inequality d overbarminust Subscript c Baseline StartFraction s Subscript d Over StartRoot n EndRoot EndFraction less thanmu Subscript dless thand overbarplust Subscript c Baseline StartFraction s Subscript d Over StartRoot n EndRoot EndFraction .

To test the effectiveness of a new drug that is reported to increase the number of hours of sleep patients get during the​ night, researchers randomly select 13 patients and record the number of hours of sleep each gets with and without the new drug.

Patient   Sleep without the drug   Sleep using the drug
1 2.6 3.5
2 4.2   6.0
3 2.2 3.6
4 3.6 4.2
5 4.2 4.9
6 2.4 3.6
7 3.3 4.8
8 3.5 6.1
9 1.9 3.7
10 4.7   5.9
11 5.2 7.1
12 5.3 6.7
13 5.2 7.0

The confidence interval is

nothingless than

Round to two decimal places.

Answer 1

Sample #1	Sample #2	difference , Di =sample1-sample2	(Di - Dbar)²
2.6	3.5	-0.900	0.2983
4.2	6	-1.800	0.1252
2.2	3.6	-1.400	0.0021
3.6	4.2	-0.600	0.7160
4.2	4.9	-0.700	0.5567
2.4	3.6	-1.200	0.0606
3.3	4.8	-1.500	0.0029
3.5	6.1	-2.600	1.3314
1.9	3.7	-1.800	0.1252
4.7	5.9	-1.200	0.0606
5.2	7.1	-1.900	0.2060
5.3	6.7	-1.400	0.0021
5.2	7	-1.800	0.1252

	sample 1	sample 2	Di	(Di - Dbar)²
sum =	48.3	67.1	-18.8	3.612

mean of difference ,    D̅ =ΣDi / n =   -1.4462
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    0.5487
-----------------

sample size ,    n =    13          
Degree of freedom, DF=   n - 1 =    12   and α =    0.1  
t-critical value =    t α/2,df =    1.7823   [excel function: =t.inv.2t(α/2,df) ]      
                  
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    0.5487          
                  
std error , SE = Sd / √n =    0.5487   / √   13   =   0.1522
margin of error, E = t*SE =    1.7823   *   0.1522   =   0.2712
                  
mean of difference ,    D̅ =   -1.446          
confidence interval is                   
Interval Lower Limit= D̅ - E =   -1.446   -   0.2712   =   -1.7174
Interval Upper Limit= D̅ + E =   -1.446   +   0.2712   =   -1.1749
                  
so, confidence interval is (   -1.72 < Dbar <   -1.17 )