Construct a 99% confidence interval for the population mean, ì. Assume the population has a normal distribution. A group of 19 randomly selected employees has a mean age of 22.4 years with a standard deviation of 3.8 years. Round to the nearest tenth.
Well, constructing a confidence interval is serious business, but I'll do my best to inject some humor into this mathematical endeavor!
First off, let's gather the information we need. We have the sample mean age of these 19 employees, which is 22.4 years, and the standard deviation, which is 3.8 years. Now let's calculate the confidence interval.
To construct a 99% confidence interval using the t-distribution, we'll need to find the critical value. This critical value depends on the sample size and the desired confidence level... but it's like trying to find Waldo in a sea of people! Well, except instead of a striped shirt, Waldo is a value on a t-distribution table.
Using my handy-dandy calculator, I found that the critical value for a 99% confidence level with 18 degrees of freedom is approximately 2.89. (No, not the pizza delivery time!)
Now, let's calculate the margin of error. The formula for the margin of error is:
Margin of Error = Critical Value * (Standard Deviation / Square Root of Sample Size)
So, plugging in the values, we get:
Margin of Error = 2.89 * (3.8 / Square Root of 19)
Now, let's compute this:
Margin of Error ≈ 2.89 * (3.8 / √19)
(Oops, I tripped on an imaginary banana peel there!)
After doing some calculations, we find that the margin of error is approximately 1.807. Now, we need to use this margin of error to build our confidence interval.
Confidence Interval = Sample Mean ± Margin of Error
Confidence Interval ≈ 22.4 ± 1.807
And finally, rounding to the nearest tenth, we have:
Confidence Interval ≈ (20.6, 24.2)
So, the 99% confidence interval for the population mean age (ì) based on this sample is somewhere between 20.6 and 24.2 years old. Just remember, this is a statistical estimation, not an invitation to shout "Happy Birthday" to your coworkers!
To construct a 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / square root of sample size)
Step 1: Calculate the standard error of the mean (SEM):
SEM = standard deviation / square root of sample size
= 3.8 / √19
≈ 0.872
Step 2: Determine the critical value for a 99% confidence interval. Since the sample size is small (n < 30), we need to use a t-distribution. The degrees of freedom is (n - 1) = (19 - 1) = 18. Consulting a t-table or calculator, we find that the critical value for a 99% confidence interval with 18 degrees of freedom is approximately 2.878.
Step 3: Calculate the confidence interval:
Confidence Interval = sample mean ± (critical value) * SEM
= 22.4 ± (2.878 * 0.872)
= 22.4 ± 2.509
Step 4: Round the confidence interval to the nearest tenth:
Confidence Interval ≈ (19.9, 24.9)
Therefore, the 99% confidence interval for the population mean is approximately (19.9, 24.9).
To construct a confidence interval, we need to use the formula:
CI = x̄ ± z * σ/√n
Where:
- CI represents the confidence interval
- x̄ is the sample mean
- z is the z-score corresponding to the desired confidence level
- σ is the population standard deviation
- n is the sample size
In this case:
- x̄ = 22.4 (sample mean)
- σ = 3.8 (population standard deviation)
- n = 19 (sample size)
First, we need to find the z-score corresponding to a 99% confidence level. Since we want a 99% confidence interval, the remaining 1% will be divided equally between the two tails of the distribution. Therefore, each tail will have an area of (1-0.99)/2 = 0.005. We can find the z-score corresponding to this area by looking it up in a standard normal distribution table or using a statistical calculator.
The z-score that corresponds to an area of 0.005 is approximately 2.58.
Now, we can plug in the values into the formula:
CI = 22.4 ± 2.58 * 3.8/√19
Calculating the expression inside the parentheses first:
2.58 * 3.8 = 9.784
Then calculating the square root of the sample size:
√19 = 4.3589
Finally, we can substitute the values into the formula:
CI = 22.4 ± 9.784/4.3589
To calculate the confidence interval, we need to evaluate the expression:
CI = 22.4 ± 2.24
Rounding to the nearest tenth, we get:
CI = (20.2, 24.6)
Therefore, the 99% confidence interval for the population mean, ì, is (20.2, 24.6).