Question

Efficiency experts study the processes used to manufacture items in order to make them as efficient as possible. One of the steps used to manufacture a metal clamp involves the drilling of three holes. In a sample of 68 clamps, the mean time to complete this step was 54.5 seconds. Assume that the population standard deviation is 6 seconds. What is the lower bound of the 98% confidence interval?

Answer 1

Given problem is about finding confidence interval for mean ($u$) which contain { lower bound, upper bound}

So in this problem we have given that,

sample size (n) = 68

mean time to complete steps = ($x^-$ ) ( or x^bar) = 54.5

population standard deviation = ($\sigma$) = 6

confidence level (c) = 0.98

So formula for confidence interval for mean ($u$) is

Vn Vn

lower bound < $u$ < upper bound

i.e lower bound =

Now to find lower bound we have all value except  

So to find , we have given confidence level (c) = 0.98

$\alpha = 1-c$ = 1-0.98 = 0.02

$\alpha/2 =0.01$

= 1-0.01 = 0.99

For that use value we need to use Z table and see above value (0.99) in Z table which will give 2.33

or Use below command in Excel

=NORMINV(0.99,0,1) = 2.33 ( Excel function to find Z value for normal distribution with mean 0 and standard deviation 1)

So   = 2.33

Now we have all value to find 98% lower bound , so we will put them in above formula,

= 54.5 - 2.33* (6/ sqrt(68)) sqrt = Square root

= 54.5 - 2.33* (6 / 8.25)

=54.5 - 1.70

=52.80

So lower bound for confidence interval is 58.8 seconds.