Question

An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this firm.

Answer 1

To find a 96% confidence interval for the population mean of all bulbs produced by the electrical firm, calculate the sample mean ± (critical value) * (standard deviation / square root of sample size).

Step-by-step explanation:

To find a 96% confidence interval for the population mean of all bulbs produced by the electrical firm, we can use the formula:

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

Since the sample size is 30 and the standard deviation is 40, we first need to find the critical value. The critical value for a 96% confidence level is 2.052, which can be found using a standard normal distribution table or a calculator.

Plugging in the values, we get:

Confidence Interval = 780 ± (2.052) * (40 / √30)

Calculating the values, we find that the 96% confidence interval for the population mean is approximately 766.5 to 793.5 hours.

Answer 2

The 96% confidence interval for the population mean of all bulbs produced by this firm is between 765 hours and 795 hours.

Step-by-step explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1-0.96)/(2) = 0.02`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`

So it is z with a pvalue of 1-0.02 = 0.98, so z = 2.055

Now, find the margin of error M as such

`M = z*(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

So

`M = 2.055*(40)/(√(30)) = 15`

The lower end of the interval is the mean subtracted by M. So 780 - 15 = 765 hours.

The upper end of the interval is M added to the mean. So 780 + 15 = 795 hours.

The 96% confidence interval for the population mean of all bulbs produced by this firm is between 765 hours and 795 hours.