3. (30 Points) A civil engineer is analyzing the compressive strength of concret

Question

3. (30 Points) A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately normally distributed with variance -1100 psi A random sample of 15 specimens has a mean compressive strength of 3355.42 psi. a) Test the hypothesis that mean compressive strength is 3600 psi with =0.01. Hypothesis: Test Statistic: Critical Value: Decision: Conclusion: b) What is the P-value? Make conclusions. c) Find a 99% two-sided CI. d) What is the -error if the true mean is 3650 psi?

Explanation / Answer

Question 3

Solution:

Part a

Here, we have to use one sample z test for population mean. The null and alternative hypotheses for this test are given as below:

Hypotheses:

Null hypothesis: H0: The mean compressive strength is 3600 psi.       

Alternative hypothesis: Ha: The mean compressive strength is different than 3600 psi.

H0: µ = 3600 versus Ha: µ 3600

This is a two tailed test.

We are given a level of significance = = 0.01

Test statistic:

Test statistic formula for this test is given as below:

Z = (Xbar - µ) / [ / sqrt(n) ]

We are given

Xbar = 3355.42

2 = 1100

= sqrt(1100) = 33.16625

n = 15

Z = (3355.42 – 3600) / [33.16625 / sqrt(15)]

Z = -28.5608

Critical value:

Lower critical value = -2.5758

Upper critical value = 2.5758

(By using z-table)

Decision:

Here, test statistic value is less than lower critical value, so we reject the null hypothesis

Conclusion:

We reject the null hypothesis that mean compressive strength is 3600 psi.          

There is sufficient evidence to conclude that mean compressive strength is different than 3600 psi.

There is insufficient evidence to conclude that mean compressive strength is 3600 psi.

Part b

By using z-table or excel,

P-value = 0.00

= 0.01

P-value <

So, we reject the null hypothesis that mean compressive strength is 3600 psi.   

There is sufficient evidence to conclude that mean compressive strength is different than 3600 psi.

There is insufficient evidence to conclude that mean compressive strength is 3600 psi.

Part c

We have to find 99% two sided confidence interval.

Confidence interval = Xbar -/+ Z*/sqrt(n)

Critical value for 99% confidence interval is 2.5758 (by using z-table).

Confidence interval = 3355.42 -/+ 2.5758*33.16625/sqrt(15)

Confidence interval = 3355.42 -/+ 2.5758* 8.5635

Confidence interval = 3355.42 -/+ 22.0581

Lower limit = 3355.42 - 22.0581 = 3333.36

Upper limit = 3355.42 + 22.0581 = 3377.48

Confidence interval = (3333.36, 3377.48)

Part d

We are given that true mean = 3650

= P(type II error) = P(µ < true mean) = P(µ<3600)

z = (3600 – 3650) / [33.16625/sqrt(15)]

z = -5.83874

P(Z<-5.83874) = 0.00000000262

= 0.00000000262

= 0.00 (approximately)

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