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.
Formula:
CI99 = mean + or - 2.575(sd divided by √n)
...where + or - 2.575 represents the 99% confidence interval using a z-table, sd = standard deviation, √ = square root, and n = sample size.
With your data:
CI99 = 22 + or - 2.575(5/√100)
Finish the calculation for your confidence interval estimate.
Formula for the second part:
n = [(z-value * sd)/E]^2
...where n = sample size, z-value will be 2.575 using a z-table to represent the 99% confidence interval, sd = 5, E = 1, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.
To compute the 99% confidence interval estimate of the mean of the population, we will use the formula:
CI99 = mean ± 2.575(sd / √n)
Using the given data, we have:
mean = 22 minutes
sd = 5 minutes
n = 100 (sample size)
Plugging these values into the formula, we get:
CI99 = 22 ± 2.575(5 / √100)
Now let's calculate:
CI99 = 22 ± 2.575(5 / 10)
= 22 ± 2.575(0.5)
= 22 ± 1.2875
Therefore, the 99% confidence interval estimate of the mean of the population is:
CI99 = (22 - 1.2875, 22 + 1.2875)
= (20.7125, 23.2875)
Now let's move on to the second part of the question, determining the necessary sample size if we want to be 99% confident and tolerate an error of 1 minute.
For this, we will use the formula:
n = [(z-value * sd) / E]^2
Given values:
z-value = 2.575 (for 99% confidence interval)
sd = 5 minutes
E = 1 minute (tolerable error)
Plugging these values into the formula, we get:
n = [(2.575 * 5) / 1]^2
= (12.875 / 1)^2
= 12.875^2
≈ 166.016
Rounding up to the next highest whole number, the necessary sample size would be approximately 167.
Therefore, to be 99% confident and tolerate an error of 1 minute, you would need a sample size of at least 167.