Computer Depot is a large store that sells and repairs computers. A random sample of 110 computer repair jobs took technicians an average of 93.2 minutes per computer. Assume that o is known to be 16.9 minutes. Find a 99% confidence interval for the population mean u for the computer repairs. Interpret the interval in the context of this problem. State what assumptions were met to use the method you selected.
For raw score, use SD, but for means, use Standard Error of the mean (SEm).
99% = mean ± 2.585 SEm
SEm = SD/√n
678 to 56.8
To find a 99% confidence interval for the population mean for computer repairs, we can use the formula:
CI = x̄ ± z * (σ/√n)
where:
CI = Confidence Interval
x̄ = Sample mean (93.2 minutes)
z = Z-score Critical Value for 99% confidence interval (we can look this up in the Z-table, it is approximately 2.576)
σ = Known population standard deviation (16.9 minutes)
n = Sample size (110)
Now, let's calculate the confidence interval:
CI = 93.2 ± 2.576 * (16.9/√110)
CI = 93.2 ± 2.576 * (16.9/√110)
CI = 93.2 ± 2.576 * 1.61
CI = 93.2 ± 4.15
Therefore, the 99% confidence interval for the population mean for computer repairs is (89.05, 97.35) minutes.
Interpretation: We are 99% confident that the true population mean for computer repair time at Computer Depot falls within the range of 89.05 minutes to 97.35 minutes. This means that if we randomly selected another set of 110 computer repair jobs and calculated the average repair time, we would expect it to be within this interval 99% of the time.
Assumptions:
1. The sample is random and representative of the population of computer repair jobs at Computer Depot.
2. The sample size is sufficiently large (n = 110).
3. The population standard deviation (σ) is known, which allows us to use the Z-distribution.
4. The data follows a normal distribution.
To find a 99% confidence interval for the population mean u for the computer repairs, we can use the formula:
Confidence Interval = sample mean ± Z * (o / sqrt(n))
Where:
- sample mean is the average repair time for the random sample (93.2 minutes)
- o is the known standard deviation (16.9 minutes)
- n is the sample size (110)
- Z is the critical value from the standard normal distribution corresponding to the desired level of confidence (99% in this case)
First, we need to find the critical value (Z) at a 99% confidence level. The area of confidence outside the interval is (100% - 99%)/2 = 0.5%. Since the standard normal distribution is symmetric, we need to find the Z-score corresponding to an area of 0.5% in both tails. By referring to a standard normal distribution table or using a statistical calculator, we can find that the Z-score is approximately 2.58.
Plugging the values into the formula:
Confidence Interval = 93.2 ± 2.58 * (16.9 / sqrt(110))
Calculating the confidence interval:
Confidence Interval = 93.2 ± 2.58 * (16.9 / sqrt(110))
Confidence Interval ≈ 93.2 ± 3.95
The 99% confidence interval for the population mean u for the computer repairs is approximately (89.25, 97.15). This means that we are 99% confident that the true average repair time for all computer repairs at Computer Depot falls between 89.25 and 97.15 minutes.
Assumptions for using this method:
1. The data was randomly sampled from the population of computer repairs.
2. The sample size is sufficiently large (n ≥ 30) for application of the Central Limit Theorem.
3. The repair times follow a normal distribution or the sample size is large enough for the Central Limit Theorem to hold.
4. The known standard deviation (o) is a valid measure of the population standard deviation.