Question

1. "Durable press" cotton fabrics are treated to improve their recovery from wrinkles after washing. "Wrinkle recovery angle" measures how well a fabric recovers from wrinkles. Higher is better. Here are data on the wrinkle recovery angle (in degrees) for the same fabric specimens discussed in the previous exercise:

Permafresh	Hylite
10	19
11	15
13	16
12	17
14	18
	14

A manufacturer wants to know how large is the difference in mean wrinkle recovery angle.

Give a 98% confidence interval for the difference in mean wrinkle recovery angle:

(  ,	)
[three decimal accuracy]	[three decimal accuracy]

2. Heart rates are determined before and 30 minutes after a Kettleball workout. It can be assumed that heart rates (bpm) are normally distributed. Use the data provided below to test to determine if average heart rates prior to the workout are significantly lower than 30 minutes after a Kettleball workout at the 0.02 level of significance. Let μ1μ1 = mean before workout.

before 60 67 71 64 75 61 72 64 64 after 58 70 72 64 79 66 76 68 68

Construct the appropriate confidence interval for the given level of significance.

(	,		)
	[three decimal accuracy]	[three decimal accuracy]

Answer 1

1)

Sample #1   ---->   1
mean of sample 1,    x̅1=   12.000
standard deviation of sample 1,   s1 =    1.581
size of sample 1,    n1=   5
      
Sample #2   ---->   2
mean of sample 2,    x̅2=   16.500
standard deviation of sample 2,   s2 =    1.871
size of sample 2,    n2=   6

Degree of freedom, DF=   n1+n2-2 =    9              
t-critical value =    t α/2 =    2.8214   (excel formula =t.inv(α/2,df)          
                      
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    1.7480              
                      
std error , SE =    Sp*√(1/n1+1/n2) =    1.0585              
margin of error, E = t*SE =    2.8214   *   1.06   =   2.9864  
                      
difference of means =    x̅1-x̅2 =    12-16.5=               -4.5000
confidence interval is                       
Interval Lower Limit=   (x̅1-x̅2) - E =    -4.5000   -   2.986   =   -7.486
Interval Upper Limit=   (x̅1-x̅2) + E =    -4.5000   +   2.986   =   -1.514

2)

SAMPLE 1	SAMPLE 2	difference , Di =sample1-sample2	(Di - Dbar)²
60	58	2	20.753
67	70	-3	0.198
71	72	-1	2.420
64	64	0	6.531
75	79	-4	2.086
61	66	-5	5.975
72	76	-4	2.086
64	68	-4	2.086
64	68	-4	2.086

sample size ,    n =    9          
Degree of freedom, DF=   n - 1 =    8   and α =    0.02  
t-critical value =    t α/2,df =    2.8965   [excel function: =t.inv.2t(α/2,df) ]      
                  
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    2.3511          
                  
std error , SE = Sd / √n =    2.3511   / √   9   =   0.7837
margin of error, E = t*SE =    2.8965   *   0.7837   =   2.2700
                  
mean of difference ,    D̅ =   -2.556          
confidence interval is                   
Interval Lower Limit= D̅ - E =   -2.556   -   2.2700   =   -4.826
Interval Upper Limit= D̅ + E =   -2.556   +   2.2700   =   -0.286