A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately Normally distributed...

A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately Normally distributed with variance ? 2 = 1000 psi2 . A random sample of ? = 12 specimens has a mean compressive strength of ?̅= 3179.5 psi. Let ? be the true mean compressive strength of all specimens from this type of concrete. (a) Test ?0: ? = 3200 against ??: ? < 3200 at ? = 0.05. Give the rejection region. (b) If the true value of ? is 3190 psi, what is the power of the test? Use ? = 0.05. (c) Compute a 2-sided 95% CI for ? the mean compressive strength. (d) How large a sample does the engineer need so that a 2-sided 95% CI for ? will have a margin of error ? of at least 15 psi?

Solutions

Expert Solution

a)
Below are the null and alternate hypothesis
H0: mu = 3200
Ha: mu < 3200

This is left tailed z - test
Critical value of test statistic is -1.6449

Here, n = 12 , xbar = 3179.5 , mu = 3200 and s = 31.6228

Test statistic,
z = (xbar - mu)/(s/sqrt(n))
z = (3179.5 - 3200)/(31.6228/(sqrt(12))
z = -2.2457

p-value = 0.0124
As p-value is less than the significance level, we reject the null hypothesis.

b)
Here, n = 12 , xbar = 3179.5 and s = 31.6228
Critical value is -1.6449
True mean = 3190

Critical value, c = xbar + (z/t)*s/sqrt(n)
c = 3179.5 + -1.6449 * 31.6228/sqrt(12)
c = 3164.4842

Beta or type II error is the probability of fail to reject the null hypothesis
P(X > 3194.5158 | mu = 3190) = 0.3104
beta = 0.3104
Power of the test is 1 - beta
Power of the test is 0.6896

c)
Two Sided Confidence Interval for 95%
Here, n = 12 , xbar = 3179.5 and s = 31.6228
z-value is 1.96

Standard Error, SE = s/sqrt(n)
SE = 31.6227766016838/sqrt(12)
SE = 9.1287

Margin of Error, ME = z*SE
ME = 1.96*9.1287
ME = 17.8923

CI = (xbar - ME, xbar + ME)
CI = (3179.5 - 17.8923 , 3179.5 + 17.8923)
CI = (3161.6077 , 3197.3923)

d)
Here, ME = 15 and sigma, s = 31.6227766016838
z-value is 1.96
sample size, n = (z*sigma/ME)^2
n = (1.96 * 31.6227/15)^2
n = 18

orchestra answered 1 year ago

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