A construction firm thinks that it is receiving 100 steel pipes with an average tensile strength of 10,000 pounds per square inch(lbs p.s.i.).This is the mean,μ.The size of the sample was n=100.The firm also knows that the population standard deviation,sigma,σ,is 400 p.s.i.The firm chooses a confidence interval of 95 %.This is equivalent to a level of significance,α,of 5 %(.05),where the null hypothesis is H0:μ0=10,000 and the alternative hypothesis is H1:μ0≠10,000.The company does not know that the actual, average tensile strength is not μ0=10,000,but μ1 =9,940.
Calculate the probability of making a Type II (β) error, the power of the Test, AND WHETHER YOU CAN CONTINUE ON AND PERFORM A TYPE I error test.
The values of sample mean X̅ for which null hypothesis is
rejected
Z = ( X̅ - µ ) / ( σ / √(n))
Critical value Z(α/2) = Z( 0.05 /2 ) = ± 1.96
1.96 = ( X̅ - 10000 ) / ( 400 / √( 100 ))
X̅1 <= 9921.6
X̅2 >= 10078.4
P ( X̅1 <= 9921.6 | µ = 9940 ) = 0.3228
P ( X̅2 >= 10078.4 | µ = 9940 ) = 0.0003
P ( Type II error ) = fail to reject null hypothesis | when H0 is false
Probability of type II error = 0.6769
Power of test is = 1 - P ( Type II error ) =
0.3231
P ( Type I error ) = reject null hypothesis | H0 is true = α
P ( Type I error ) = 0.05
A construction firm thinks that it is receiving 100 steel pipes with an average tensile strength...
An engineer studying the tensile strength of a composite material knows that tensile strength is approximately normally distributed with σ = 60 psi. A random sample of 20 specimens has a mean tensile strength of 3450 psi. (a) Test the hypothesis that the mean tensile strength is 3500 psi, using α = 0.01 (b) What is the smallest level of significance at which you would be willing to reject the null hypothesis? (c) What is the β error for the...