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 95% confidence interval for the population mean of all bulbs supplied by this company. Give the lower boundary.

Question

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 95% confidence interval for the population mean of all bulbs supplied by this company. Give the lower boundary.

Answer 1

To calculate the 95% confidence interval for the population mean of all bulbs supplied by the company, we can use the formula:

Confidence interval = sample mean ± (critical value * standard deviation / sqrt(sample size))

First, we need to determine the critical value. In a normal distribution, the critical value for a 95% confidence interval is approximately 1.96.

Next, we plug in the given values into the formula:

Confidence interval = 780 ± (1.96 * 40 / sqrt(36))

Simplifying this expression:

Confidence interval = 780 ± (1.96 * 40 / 6)

Confidence interval = 780 ± 27.87

To find the lower boundary of the confidence interval, we subtract 27.87 from the sample mean:

Lower boundary = 780 - 27.87

Lower boundary ≈ 752.13

Therefore, the lower boundary of the 95% confidence interval for the population mean of all bulbs supplied by the company is approximately 752.13.

Answer 2

To find a 95% confidence interval for the population mean, we can use the formula:

Confidence Interval = sample mean ± (critical value) * (standard deviation / square root of sample size)

First, let's calculate the critical value for a 95% confidence level. Since we have a large sample size (n ≥ 30), we can use the Z-value for a 95% confidence level, which is approximately 1.96.

Next, let's plug in the values into the formula:

Sample mean (x̄) = 780 hours
Standard deviation (σ) = 40 hours
Sample size (n) = 36

Confidence Interval = 780 ± (1.96) * (40 / √36)

To calculate the square root of the sample size (√36), we know that the square root of 36 is 6.

Confidence Interval = 780 ± (1.96) * (40 / 6)

Now, calculate the value inside the parentheses:

Confidence Interval = 780 ± (1.96) * (6.67)

Multiply 1.96 by 6.67:

Confidence Interval = 780 ± 13.05

Finally, calculate the lower boundary of the confidence interval:

Lower Boundary = 780 - 13.05

Lower Boundary = 766.95

Therefore, the lower boundary of the 95% confidence interval for the population mean of all bulbs supplied by this company is approximately 766.95 hours.