Question

A random sample of 42 mechanical gears has been collected, and the measurements of
Gear-Diameter (mm) are found below:
130 131 132 130.5 130 130.2 130.7 130 131.2 133
130 130 131 132 133 133 132 131 312.5 131.5
131.2 132.5 130.2 129.5 131 130 130.2 131.5 131.4 131.6
130.6 130.5 131.1 130 13 130 131 130 130 132
131.2 132
(a) Construct a 94% two-bounds confidence interval on the diameter.
(b) Construct a 94% lower confidence bound on the diameter.

please solve manually

Answer 1

A random sample of 42 mechanical gears is given below

130 131 132 130.5 130 130.2 130.7 130 131.2 133
130 130 131 132 133 133 132 131 312.5 131.5
131.2 132.5 130.2 129.5 131 130 130.2 131.5 131.4 131.6
130.6 130.5 131.1 130 13 130 131 130 130 132
131.2 132

[ Note that one observation is " 13 " which looks like incorrect entry , but due to no evidence we will continue calculation with considering "13" as correct entry ]

Here n = 42 ( sample size )

Now we will calculate sample mean and sample variance s²

=

   = ( 130 +131 +132 +130.5 +130 + ........ + 130 +132 +131.2 +132 ) /42

   = 5565.1 / 42

= 132.5024

And

s² =

= { (130-132.5024)² +(131-132.5024)² +(132-132.5024)² +....... + (131.2-132.5024)² +(132-132.5024)² } / (42-1)

s² = 46806.77 / 41

s² = 1141.629

Thus sample standard deviation s = = 33.78799

So we have

= 132.5024     ,      s = 33.78799

(a) Construct a 94% two-bounds confidence interval on the diameter.

94% two-bounds confidence interval CI is given by

CI = {   - * , - * }

where is t-distributed with n-1 = 42-1 = 41 degree of freedom (DF) and = 0.06 { for 94% confidence }

Now at = 0.06 with DF = 41 , can be obtained from statistical book or from any software like R/Excel

From R

> qt(1-0.06/2,df=41)            
[1] 1.934283

Hence = 1.934283

Thus

CI = {   - * , + * }

CI = {   132.5024 - 1.934283 * , 132.5024 + 1.934283 * }

CI = { 122.4178 , 142.587 }                         [ after calculation ]

Hence 94% two-bounds confidence interval on the diameter is { 122.4178 , 142.587 }   

b) Construct a 94% lower confidence bounds on the diameter.

94% lower confidence bound LB is given by

LB =    - *   

Now is t-distributed at = 0.06 with DF = 41

Calculation from R , since in can't find accurate value at df=41 and = 0.06 in statistical book

From R

> qt(1-0.06,df=41)
[1] 1.587865

Hence   = 1.587865

Thus

LB = - *

LB = 132.5024 - 1.587865 *

LB = {   132.5024 - 8.278494

LB = 124.2239  

{ CI = { 124.2239 , Inf } or > 124.2239 }

Hence 94% lower confidence bounds on the diameter is LB = 124.2239