Question

Q1: A cyclist collected the below data on RPMs during his ride:

58 56 50 60 55 50 52 53 59 53 55 57 57 63 48
55 47 47 52 54 53 55 46 55 56 50 55 50 57 54
68 50 62 64 53 50 64 59 61 55 53 41 57 53 50

w. Create a one-sided interval that will contain 99% of the RPMs, with 95% confidence.

x. Create a two-sided 90% interval on the RPM collected in the next test.

y. Create a 90% confidence interval on the standard deviation of the collected RPMs.

z. Create a two-sided interval that is 95% confident on containing the RPMs of all of the next five tests.

Answer 1

x)

Level of Significance ,    α =    0.1          
degree of freedom=   DF=n-1=   44          
't value='   tα/2=   1.680   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   5.277/√45=   0.7866          
margin of error , E=t*SE =   1.6802   *   0.7866   =   1.3217
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    54.49   -   1.321744   =   53.1671
Interval Upper Limit = x̅ + E =    54.49   -   1.321744   =   55.8106
90%   confidence interval is (   53.17   < µ <   55.81   )

y)

Sample Size,   n=   45
Sample Standard Deviation,   s=   5.2770
Confidence Level,   CL=   0.90
      
      
Degrees of Freedom,   DF=n-1 =    44
alpha,   α=1-CL=   0.1
alpha/2 ,   α/2=   0.05
Lower Chi-Square Value=   χ²1-α/2 =   29.787
Upper Chi-Square Value=   χ²α/2 =   60.481

confidence interval for std dev is       
lower bound= √[(n-1)s²/χ²α/2] =   √(44*5.277² / 60.4809)=   4.5009
      
      
upper bound= √[(n-1)s²/χ²1-α/2] =   √(44*5.277² / 29.7875)=   6.4135