To find a 90% confidence interval for the population mean of the light bulbs' life expectancy, we will use the formula for the confidence interval for the mean when the population standard deviation is known. The formula is:
\[ \bar{x} \pm z \times \left( \frac{\sigma}{\sqrt{n}} \right) \]
where:
- \(\bar{x}\) is the sample mean
- \(z\) is the z-value corresponding to the desired confidence level
- \(\sigma\) is the population standard deviation
- \(n\) is the sample size
Given Data:
- Sample mean \(\bar{x} = 780\) hours
- Population standard deviation \(\sigma = 40\) hours
- Sample size \(n = 36\)
Step 1: Find the z-value for a 90% confidence level.
For a 90% confidence interval, the area in each tail is \( (1 - 0.90) / 2 = 0.05\). Therefore, we need to find the z-value for \(0.95\) (since \(1 - 0.05 = 0.95\)).
Using a standard normal distribution table or a calculator, we find:
\[ z \approx 1.645 \]
Step 2: Calculate the standard error.
The standard error (SE) of the sample mean is given by:
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{40}{\sqrt{36}} = \frac{40}{6} \approx 6.67 \]
Step 3: Calculate the margin of error.
The margin of error (ME) is calculated as:
\[ ME = z \times SE = 1.645 \times 6.67 \approx 10.97 \]
Step 4: Construct the confidence interval.
Now we can calculate the confidence interval:
\[ \text{Confidence Interval} = \bar{x} \pm ME = 780 \pm 10.97 \]
This gives us:
\[ (780 - 10.97, 780 + 10.97) = (769.03, 790.97) \]
Final Result:
The 90% confidence interval for the population mean of all light bulbs supplied by this company is approximately:
\[ (769.03, 790.97) \]