# Tests of Hypotheses math statistics

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 = 56. Let μ denote the true average compressive strength
B)Let
X
denote the sample average compressive strength for n = 12 randomly selected specimens. Consider the test procedure with test statistic
X
and rejection region x ≥ 1331.26.

mean of the test statistic KN/m2

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standard deviation of the test statistic KN/m2

Using the test procedure of part (b), 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.)
C)
What is the probability distribution of the test statistic when μ = 1350?
mean of the test statistic KN/m2

standard deviation of the test statistic KN/m2

Using the test procedure of part (b), 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.)

(d) How would you change the test procedure of part (b) to obtain a test with significance level 0.05? (Round your answer to two decimal places.)
Replace 1331.26 KN/m2 with KN/m2.

What impact would this change have on the error probability of part (c)? (Round your answer to four decimal places.)

The probability that the mixture will be judged unsatisfactory when in fact μ = 1350 will change to .

(e) Consider the standardized test statistic
Z = (X − 1300)/(σ/

). What are the values of Z corresponding to the rejection region of part (b)? (Round your answer to two decimal places.)
z ≥

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design.

(c) What is the significance level for the appropriate region of part (b)? (Round your answer to four decimal places.)

How would you change the region to obtain a test with α = 0.001? (Round your answer to one decimal place.)
Replace the limit(s) in the region with .

(d) What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the appropriate region from part (b) is used? (Round your answer to four decimal places.)

(e) Let
Z = (X − 120)/(σ/

n

). What is the significance level for the rejection region {z: z ≤ −2.39}? For the region {z: z ≤ −2.69} (Round your answers to four decimal places.)

for
{z: z ≤ −2.39

for
{z: z ≤ −2.69}

Let the test statistic Z have a standard normal distribution when H0 is true. Give the significance level for each of the following situations. (Round your answers to four decimal places.)
(a) Ha: μ > μ0, rejection region
z ≥ 1.71

(b) Ha: μ [removed] μ0, df = 18, rejection region t ≥ 3.610

(b)
Ha: μ < μ0, n = 28, rejection region t ≤ −3.690

(c)
Ha: μ ≠ μ0, n = 33, rejection region t ≥ 2.037 or t ≤ −2.037

A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times, and determines that 14 of the plates have blistered.

State the rejection region(s). If the critical region is one-sided, enter NONE for the unused region. Round your answers to two decimal places.

z ≤

z ≥

Compute the test statistic value. Round your answer to two decimal places.
z =

. .
(b) If it is really the case that 15% of all plates blister under these circumstances and a sample size 100 is used, how likely is it that the null hypothesis of part (a) will not be rejected by the 0.05 test? (Round your answer to four decimal places.)

If it is really the case that 15% of all plates blister under these circumstances and a sample size 200 is used, how likely is it that the null hypothesis of part (a) will not be rejected by the 0.05 test? (Round your answer to four decimal places.)

(c) How many plates would have to be tested to have β(0.15) = 0.10 for the test of part (a)? (Round your answer up to the next whole number.)
plates

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