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 = 60. Let μ denote the true average compressive strength.

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

1. H₀: μ = 1300
H_a: μ ≠ 1300

2. H₀: μ > 1300
H_a: μ = 1300    

3. H₀: μ < 1300
H_a: μ = 1300

4. H₀: μ = 1300
H_a: μ > 1300

5. H₀: μ = 1300
H_a: μ < 1300

(b) Let X denote the sample average compressive strength for n = 12 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? (Choose of the following)

1. The test statistic has a binomial distribution.

2. The test statistic has a gamma distribution.    

3. The test statistic has a normal distribution.

4. The test statistic has an exponential distribution.

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? (Choose of the following)

1. reject H₀

_2. do not reject H₀    

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

1. The test statistic has a binomial distribution.

2. The test statistic has a gamma distribution.    

3. The test statistic has an exponential distribution.

4. The test statistic has a normal distribution.

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

mean		KN/m2
standard deviation		KN/m2

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

(a)

(b) The test statistic has Normal distribution.

As sampling distribution of sample mean follow Normal distribution.

Test statistic

= 2.31

P value = 0.0105

Since P value > 0.01 ,

do not reject H0.

(c) the test statistic has a normal distribution

Mean of sampling distribution of sample mean = population mean = 1350

Standard deviation of sampling distribution of sample mean =

= 17.321

For

zc= 2.33

P(Type 2 error) =P ( X < 1340.36 I

=

= P(z < -0.56)

= 0.2877