Question

In: Statistics and Probability

mean 94.6129 n= 31 standard deviation is 45.0612 find the 95% and 99% confidence intervals of...

mean 94.6129
n= 31
standard deviation is 45.0612

find the 95% and 99% confidence intervals of the standard devaition

find the 95% and 99% confidence interval for population mean


data set
10
19
35
39
41
56
67
67
69
73
75
78
81
82
83
83
84
91
92
94
123
125
137
138
140
142
150
152
161
163
183

Solutions

Expert Solution

sample std dev ,    s =    45.0612
Sample Size ,   n =    31
Sample Mean,    x̅ =   94.6129
      
degree of freedom=   DF=n-1=   30
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95% confidence interval for population mean

α = 1-0.95 = 0.05

't value='   tα/2=   2.0423   [Excel formula =t.inv(α/2,df) ]
          
Standard Error , SE =   s/√n =   8.0932  
margin of error ,   E=t*SE =   16.529  
confidence interval is           
Interval Lower Limit=   x̅ - E =    78.08  
Interval Upper Limit=   x̅ + E =    111.14  

-----------------------------------------------

99% confidence interval for population mean

α=1-0.99 = 0.01

't value='   tα/2=   2.7500   [Excel formula =t.inv(α/2,df) ]
          
Standard Error , SE =   s/√n =   8.0932  
margin of error ,   E=t*SE =   22.256  
confidence interval is           
Interval Lower Limit=   x̅ - E =    72.36  
Interval Upper Limit=   x̅ + E =    116.87  

-------------------------------------------------------------------

95% confidence intervals of the standard devaition

Sample Size,   n=   31
Sample Standard Deviation,   s=   45.0612
Confidence Level,   CL= 95%
      
      
Degrees of Freedom,   DF=n-1 =    30
alpha,   α=1-CL=   0.05
alpha/2 ,   α/2=   0.025

from chi square table,
Lower Chi-Square Value=   χ²1-α/2 =   16.7908
Upper Chi-Square Value=   χ²α/2 =   46.9792

95% confidence interval for std dev is      
lower bound= √[ (n-1)s²/χ²α/2 ] =   36.0089
      
      
upper bound= √[ (n-1)s²/χ²1-α/2 ] = 60.2321

-------------------------------------------------------------------------------------


      

99% confidence intervals of the standard devaition

Sample Size,   n=   31  
Sample Standard Deviation,   s=   45.0612  
Confidence Level,   CL= 99%
          
          
Degrees of Freedom,   DF=n-1 =    30


alpha,   α=1-CL=   0.01  


alpha/2 ,   α/2=   0.005  
Lower Chi-Square Value=     χ²1-α/2 =   13.7867  
Upper Chi-Square Value=   χ²α/2 =   53.6720  
          
confidence interval for std dev  is          
  
           lower bound= √[ (n-1)s²/χ²α/2 ] =   33.6891
      
      
upper bound= √[ (n-1)s²/χ²1-α/2 ] = 66.4711
  


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