A random sample of 42 college graduates revealed that they worked an average of 5.5 years on the job before being promoted. The sample standard deviation was 1.1 years. Using the .99 degree of confidence, what is the confidence interval for the population mean?
99% = mean ± 2.54SD = ?
5.06 and 5.94
To calculate the confidence interval for the population mean, we will use the formula:
Confidence Interval = Sample Mean ± (Confidence Level * Standard Error)
1. Calculate the standard error (SE) using the formula:
SE = Sample Standard Deviation / √(Sample Size)
SE = 1.1 / √(42)
SE ≈ 0.169
2. Determine the critical value for a 99% confidence level. Since the sample size is large (n > 30), we can use the Z-table. The critical value for a 99% confidence level is 2.576.
3. Plug in the values into the confidence interval formula:
Confidence Interval = 5.5 ± (2.576 * 0.169)
4. Calculate the confidence interval:
Confidence Interval = 5.5 ± 0.434
Confidence Interval ≈ (5.066, 5.934)
Therefore, the confidence interval for the population mean is approximately (5.066, 5.934) years.
To calculate the confidence interval for the population mean, we need to use the formula:
Confidence interval = sample mean ± (critical value * standard deviation / √sample size)
In this case, the sample mean is 5.5 years, the sample standard deviation is 1.1 years, and the sample size is 42.
First, we need to find the critical value associated with the .99 degree of confidence. Since the sample size is large (42), we can use the Z-distribution to find the critical value.
The z-value for a .99 level of confidence means we want to leave 1% in the tails of the distribution, so we need to locate the z-value that corresponds to the area of 0.01 in the upper tail.
Using a standard normal distribution table or a calculator, we can find that the z-value is approximately 2.33.
Now, we can calculate the confidence interval:
Confidence interval = 5.5 ± (2.33 * 1.1 / √42)
To calculate the standard error (standard deviation / √sample size):
Standard error = 1.1 / √42 ≈ 0.17
Confidence interval = 5.5 ± (2.33 * 0.17)
Confidence interval ≈ 5.5 ± 0.3961
The confidence interval for the population mean is approximately (5.104, 5.896) years, where 5.104 is the lower bound and 5.896 is the upper bound. This means we are 99% confident that the true population mean falls within this range.