It is known that the amount of time needed to change the oil in a car is normally distributed with a standard deviation of 5 minutes. A random sample of 100 oil changes yielded a sample mean of 22 minutes. Compute the 99% confidence interval estimate of the mean of the population.
Also determine the necessary sample size if you wish to be 99% confident and can tolerate an error of 1 minute.
Part 3 - With a sample size not exceeding 150 but everything else staying the same, what confidence level can be achieved?
95
96
97
98
still 99
This was answered in a previous post. For part 3, you can probably still use the same confidence interval.
To compute the 99% confidence interval estimate of the mean of the population, we can use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)
1. Compute the margin of error:
The critical value is the z-score corresponding to the desired confidence level. In this case, since we want a 99% confidence level, we need to find the z-score that leaves 1% in the tails of the standard normal distribution. Using a z-table or a statistical software, we find that the critical value is approximately 2.576.
The standard deviation is given as 5 minutes.
The square root of the sample size is the square root of 100, which is 10.
So the margin of error is: 2.576 * 5 / 10 = 1.288.
2. Calculate the confidence interval:
The sample mean is given as 22 minutes.
The lower bound of the confidence interval is: 22 - 1.288 = 20.712.
The upper bound of the confidence interval is: 22 + 1.288 = 23.288.
Therefore, the 99% confidence interval estimate of the mean of the population is (20.712, 23.288).
For determining the necessary sample size to be 99% confident with a tolerance of 1 minute, we can use the following formula:
n = (z * sigma / E)^2
Where:
n = sample size
z = critical value (corresponding to the desired confidence level, in this case, 2.576 for a 99% confidence level)
sigma = standard deviation (given as 5 minutes)
E = desired margin of error (given as 1 minute)
Plugging in the values:
n = (2.576 * 5 / 1)^2
n = 13.504^2
n ≈ 182.31
Therefore, the necessary sample size to be 99% confident and tolerate an error of 1 minute is approximately 182.
For part 3, if the sample size is not exceeding 150 but everything else stays the same, we can still use the same confidence interval formula. The confidence level is determined by the critical value, which remains the same for a given level of confidence. Therefore, we can still achieve a 99% confidence level.