Light bulbs supplied by a certain company have a length life that is approximately normally distributed, with the standard deviation of 40 hours. If a random sample of 36 bulbs has an average life of 780 hours, find a 90% confidence interval for the population mean of all bulbs supplied by this company. Give the lower limit.
To find the confidence interval, we need to find the margin of error, which depends on the standard deviation of the population, the sample size, and the desired level of confidence.
First, we calculate the standard error, which is the standard deviation of the sample mean:
standard error = standard deviation / square root of sample size
= 40 / √36
= 40 / 6
= 6.67 (rounded to two decimal places)
Next, we determine the margin of error, which is the product of the standard error and the critical value. The critical value depends on the desired level of confidence. For a 90% confidence interval, the critical value is approximately 1.645.
margin of error = critical value * standard error
= 1.645 * 6.67
= 10.94 (rounded to two decimal places)
Finally, the confidence interval for the population mean is the sample mean plus or minus the margin of error:
confidence interval = sample mean ± margin of error
= 780 ± 10.94
Since we are looking for the lower limit of the confidence interval, we subtract the margin of error from the sample mean:
lower limit = sample mean - margin of error
= 780 - 10.94
= 769.06 (rounded to two decimal places)
Therefore, the lower limit of the 90% confidence interval for the population mean of all bulbs supplied by this company is approximately 769.06.
To find the 90% confidence interval for the population mean, we will use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / √n)
Where:
- Sample mean: 780 hours
- Standard deviation: 40 hours
- n (sample size): 36
- Critical value: To find the critical value, we will use the Z-score for a 90% confidence level (since the sample size is large).
The Z-score for the 90% confidence level is 1.645.
Now, let's calculate the confidence interval:
Confidence Interval = 780 ± (1.645 * 40 / √36)
Confidence Interval = 780 ± (1.645 * 40 / 6)
Confidence Interval = 780 ± 10.966
To find the lower limit of the confidence interval, subtract the result from the sample mean:
Lower Limit = 780 - 10.966
Lower Limit ≈ 769.034
Therefore, the lower limit of the 90% confidence interval for the population mean is approximately 769.034.