A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately normally distributed with variance σ2 = 1000 psi2. A random sample of 12 specimens has a mean compressive strength of
(a) Test the hypothesis that mean compressive strength is 3500 psi. Use a fixed-level test with α = 0.01.
(b) What is the smallest level of significance at which you would be willing to reject the null hypothesis?
(c) Construct a 95% two-sided CI on mean compressive strength.
(d) Construct a 99% two-sided CI on mean compressive strength. Compare the width of this confidence interval with the width of the one found in part (c). Comment.
The Correct Answer and Explanation is :
To address the given statistical problems, we need the sample mean ((\bar{X})) of the 12 concrete specimens. However, since it is not provided, we will express the solutions in terms of (\bar{X}).
(a) Hypothesis Testing at (\alpha = 0.01):
We aim to test the null hypothesis (H_0: \mu = 3500) psi against the alternative (H_1: \mu \neq 3500) psi. Given the population variance (\sigma^2 = 1000) psi² and sample size (n = 12), the standard error (SE) is:
[
\text{SE} = \sqrt{\frac{\sigma^2}{n}} = \sqrt{\frac{1000}{12}} \approx 9.13 \text{ psi}
]
The test statistic (z) is calculated as:
[
z = \frac{\bar{X} – 3500}{\text{SE}} = \frac{\bar{X} – 3500}{9.13}
]
For a two-tailed test at (\alpha = 0.01), the critical z-values are (\pm 2.576). If (z) falls outside this range, we reject (H_0). Thus, we reject (H_0) if:
[
|\bar{X} – 3500| > 2.576 \times 9.13 \approx 23.53 \text{ psi}
]
(b) Smallest Significance Level ((p)-value):
The (p)-value is calculated as:
[
p\text{-value} = 2 \times P(Z > |z|)
]
Where (Z) follows the standard normal distribution. The smallest significance level at which (H_0) would be rejected corresponds to this (p)-value.
(c) 95% Confidence Interval (CI) for (\mu):
A 95% CI for the mean is given by:
[
\bar{X} \pm z_{\alpha/2} \times \text{SE} = \bar{X} \pm 1.96 \times 9.13
]
This simplifies to:
[
\bar{X} \pm 17.89 \text{ psi}
]
(d) 99% Confidence Interval (CI) for (\mu):
A 99% CI is:
[
\bar{X} \pm z_{\alpha/2} \times \text{SE} = \bar{X} \pm 2.576 \times 9.13
]
This simplifies to:
[
\bar{X} \pm 23.53 \text{ psi}
]
Comparison of CI Widths:
The width of the 95% CI is (2 \times 17.89 = 35.78) psi, while the 99% CI width is (2 \times 23.53 = 47.06) psi. The 99% CI is wider, reflecting increased uncertainty due to the higher confidence level. This trade-off between confidence level and interval precision is typical in statistical analyses.
Conclusion:
Without the sample mean, we cannot compute exact numerical results. However, the provided formulas allow for substitution of (\bar{X}) to obtain specific outcomes.