Question

A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m². The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for specimens of this mixture is normally distributed with σ = 65. Let μ denote the true average compressive strength.

(a) What are the appropriate null and alternative hypotheses?

H₀: μ = 1300 H_a: μ ≠ 1300

H₀: μ < 1300 H_a: μ = 1300    

H₀: μ = 1300 H_a: μ < 1300

H₀: μ > 1300 H_a: μ = 1300

H₀: μ = 1300 H_a: μ > 1300

(b) Let

X denote the sample average compressive strength for n = 11 randomly selected specimens. Consider the test procedure with test statistic X itself (not standardized). What is the probability distribution of the test statistic when H₀ is true?

If X = 1340, find the P-value. (Round your answer to four decimal places.)
P-value =  

Should H₀ be rejected using a significance level of 0.01?

  

(c) What is the probability distribution of the test statistic when μ = 1350?

State the mean and standard deviation of the test statistic. (Round your standard deviation to three decimal places.)

For a test with α = 0.01, what is the probability that the mixture will be judged unsatisfactory when in fact μ = 1350 (a type II error)? (Round your answer to four decimal places.)

Answer 1

Ans:

a)

H0: μ = 1300 Ha: μ > 1300

b)

X-bar has Normal distribution with mean=1300 and standard deviation=65/sqrt(11)=19.598

Test statistic:

z=(1340-1300)/(65/sqrt(11))

z=2.041

P-value=P(z>2.041)=0.0206

As,p-value>0.01,we do not reject the null hypothesis.

c)

mean=1350

standard deviation=65/sqrt(11)=19.598

sample mean cut off=1300+2.326*(65/sqrt(11))=1345.586

when true mean=1350

z=(1345.586-1350)/(65/sqrt(11))

z=-0.225

P(type II error)=P(z<-0.225)=0.4109