2. An electrical firm manufactures light bulbs that have a length of life that is normally distributed with a standard deviation of 40 hours. A random sample of 30 bulbs has an average life of 780 hours.
a. Find a 96% confidence interval for the population mean of all bulbs produced by this firm.
b. How large a sample is needed if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean?
3. A random sample of 8 cigarettes of a certain brand has an average nicotine content of 3.5 milligrams and a standard deviation of 0.9 milligrams. Construct a 99% confidence interval for the true average nicotine content of this particular brand of cigarettes, assuming normal distribution.
a. To find a 96% confidence interval for the population mean of all bulbs produced by this firm, we can use the formula:
Confidence Interval = Sample Mean ± (Z * (Standard Deviation / sqrt(sample size)))
First, we need to find the critical value, Z, for a 96% confidence level. The Z-score corresponds to the area between the mean and the desired confidence level on a standard normal distribution. In this case, the confidence level is 96%, so the area outside of the confidence interval is (100% - 96%) / 2 = 2% on each tail.
Using a standard normal distribution table or a calculator, we find that the Z-score for a 2% area under each tail is approximately 2.05.
Next, we substitute the known values into the formula:
Sample Mean = 780 hours
Standard Deviation = 40 hours
Sample Size = 30
Confidence Interval = 780 ± (2.05 * (40 / sqrt(30)))
Now, calculate the lower and upper bounds of the confidence interval:
Lower Bound = 780 - (2.05 * (40 / sqrt(30)))
Upper Bound = 780 + (2.05 * (40 / sqrt(30)))
The 96% confidence interval for the population mean of all bulbs produced by this firm is [Lower Bound, Upper Bound].
b. To determine how large a sample is needed to be 96% confident that the sample mean will be within 10 hours of the true mean, we can use the formula:
Required Sample Size = (Z * (Standard Deviation / Desired Margin of Error))^2
We need to find the critical value, Z, for a 96% confidence level, just like in part (a). The desired margin of error is 10 hours.
Substituting the known values into the formula, we get:
Desired Margin of Error = 10 hours
Standard Deviation = 40 hours
Z = critical value for a 96% confidence level
Required Sample Size = (Z * (40 / 10))^2
Calculate the square of the critical value and then divide it by the square of the desired margin of error:
Required Sample Size = (Z^2 * (40^2 / 10^2))
Thus, the required sample size is equal to the value obtained above.