Question

The weight of a product is measured in pounds. A sample of 50 units is taken from a recent production. The sample yielded ¯y= 75 lb, and we know that LaTeX: \sigmaσ2= 100 lb. Calculate a 90 percent confidence interval for LaTeX: \text{μ}

Answer 1

90 percent confidence interval = [72.674 ,77.326]

Explanation:

We are given that weight of a product is measured in pounds.

A random sample of 50 units is taken from a recent production. The sample yielded y bar = 75 lb, and we know that
`\sigma^(2)` = 100 lb.

The Pivotal quantity for 9% confidence interval is given by;

`(Ybar - \mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where, Y bar = sample mean = 75

`\sigma` = population standard deviation = 10

n = sample size = 50

So, 90% confidence interval for population mean, is given by;

P(-1.6449 < N(0,1) < 1.6449) = 0.90

P(-1.6449 <
`(Ybar - \mu)/((\sigma)/(√(n) ) )` < 1.6449) = 0.90

P(-1.6449 *
`{(\sigma)/(√(n) )` <
`{Ybar - \mu}` < 1.6449 *
`{(\sigma)/(√(n) )` ) = 0.90

P(Y bar - 1.6449 *
`{(\sigma)/(√(n) )` <
`\mu` < Y bar + 1.6449 *
`{(\sigma)/(√(n) ) }` ) = 0.90

90% confidence interval for
`\mu` = [ Y bar - 1.6449 *
`{(\sigma)/(√(n) )` , Y bar + 1.6449 *
`{(\sigma)/(√(n) )` ]

= [ 75 - 1.6449 *
`{(10)/(√(50) )` , 75 + 1.6449 *
`{(10)/(√(50) )` ]

= [ 72.674 , 77.326 ]

Therefore, 90% confidence interval for population mean is [72.674 ,77.326] .