Question

Use the given degree of confidence and sample data to construct a confidence interval for the population means. A random sample of 15 light bulbs had a mean life of 548 hours and a sample standard deviation of 29 hours. Find a 90 percent confidence interval for the mean life, μ, of all light bulbs of this type.

Answer 1

Conditions

Use the given degree of confidence and sample data to construct a confidence interval for the population means. A random sample of 15 light bulbs had a mean life of 548 hours and a sample standard deviation of 29 hours. Find a 90 percent confidence interval for the mean life, $\mu$, of all light bulbs of this type.

Solution

$$0.9 = P(-1.65 \leq \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \leq 1.65)$$

Where:

1.65 – the value of cumulative distribution function at a point $p=0.9$

$\bar{X}$ – sample mean

$\mu$ – distribution mean

$\sigma$ – standard deviation

$n$ – sample size.

The confidence interval is:

$$\left(\bar{X} - 1.65 \frac{\sigma}{\sqrt{n}} \leq \mu \leq \bar{X} + 1.65 \frac{\sigma}{\sqrt{n}}\right) = \left(548 - 1.65 \frac{29}{\sqrt{15}} \leq \mu \leq 548 + 1.65 \frac{29}{\sqrt{15}}\right)$$ $$535,64518 \leq \mu \leq 560,35482$$