Question

1. application a large water pipeline is being constructed to link a town with a fresh water aquifer. a construction supervisor measured the diameters of 40 pipe segments and found that the mean diameter was 25.5 cm. in the past, pipe manufactured by the same company have had a standard deviation of 7 mm. determine a 95% confidence interval for the mean diameter of the pipe segments.

Explanation:

Identify given parameters

We extract the parameters from the problem statement:

• Sample size \(n = 40\)
• Sample mean \(\bar{x} = 25.5\text{ cm}\)
• Population standard deviation \(\sigma = 7\text{ mm} = 0.7\text{ cm}\)
• Confidence level is \(95\%\)

Find critical z-value

For a \(95\%\) confidence level:

• The significance level \(\alpha = 0.05\)
• The critical value \(z_{\alpha/2} = z_{0.025} \approx 1.96\)

Calculate standard error

The standard error of the mean is:

$$ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.7}{\sqrt{40}} \approx 0.1107\text{ cm} $$

Calculate margin of error

The margin of error \(E\) is:

$$ E = z_{\alpha/2} \times SE = 1.96 \times 0.1107 \approx 0.217\text{ cm} $$

Determine confidence interval

We calculate the lower and upper bounds:

• Lower bound: \(\bar{x} - E = 25.5 - 0.217 = 25.283\text{ cm}\)
• Upper bound: \(\bar{x} + E = 25.5 + 0.217 = 25.717\text{ cm}\)

Answer:

The \(95\%\) confidence interval for the mean diameter of the pipe segments is approximately \([25.28\text{ cm}, 25.72\text{ cm}]\).