The following sample of lengths was taken from 8 fluorescent light bulbs off the assembly line. Construct the 98% confidence interval for the population variance for all fluorescent light bulbs that come off the assembly line. Round your answers to two decimal places. 10.5,10.6,10.4,11.0,10.7,10.9,11.2, 11.3

(13.53, 8.12)

Step-by-step explanation: We are to construct a 98% confidence interval for the length of fluorescent light bulbs.

The formulae for Constructing the confidence interval is given below as

u = x + tα/2× (s/√n) For the upper limit

u = x - tα/2× (s/√n) For the lower limit

x = sample mean

s = sample standard deviation

n = sample size = 10

tα/2 = critical value of a 2 tailed t test which can be gotten from a t distribution table by checking the degree of freedom (8-1 =7) on the table against the level of significance which is 2% ( 100% - 98%).

By doing so, we have our tα/2 as 2.998

We are using a t test to get the value of our critical value (tα/2) because sample size is lesser than 30 ( n =8) and population standard deviation is unknown.

Below is data set of the length of each fluorescent bulb.

10.5,10.6,10.4,11.0,10.7,10.9,11.2,11.3.

We have to get the mean (x) and standard deviation (s) ourselves.

Mean = Σx/n

Mean = 10.5 + 10.6 + 10.4 + 11.0 + 10.7 + 10.9 + 11.2 + 11.3 / 8

Mean = 86.6 / 8 = 10.825.

s² = {Σx² - (Σx)²/n} / n - 1

Σx² = 10.5² + 10.6² + 10.4² + 11.0² + 10.7² + 10.9² + 11.2² + 11.3² = 938.2

s² = {938.2 - (86.6)²/8}/7

s² = {983.2 - 937.445}/ 7

s² = 45.755/7

s² = 6.536

s = √6.536

s = 2.557.

Let us now substitute our parameters to get our confidence interval.

For upper limit

u = x + tα/2× (s/√n)

u = 10.825 + 2.998 ( 2.557/√8)

u = 10.825 + 2.998 (0.904)

u = 10.825 + 2.71

u = 13.53.

For lower limit

u = x - tα/2× (s/√n)

u = 10.825 - 2.998 ( 2.557/√8)

u = 10.825 - 2.998 (0.904)

u = 10.825 - 2.71

u = 8.12

Hence the 98% confidence interval for mean length of fluorescent light bulbs is (13.53, 8.12)

you can't solve x because it is 2 linear equation and you only have one equation