Question
From a sample of 64 motorists working in Cape Town, the sample average time that it takes them to drive to work is 36 minutes. It is known that the population standard deviation is 20 minutes. Construct a 99% confidence interval for the actual average time it takes all motorists in Cape Town to drive to work. What is the critical value of the interval?
2.576
2.807
3.090
1.96
2.576
2.807
3.090
1.96
Answers
GPT-4o mini
To construct a 99% confidence interval for the average time it takes all motorists in Cape Town to drive to work, we need to use the formula for the confidence interval for the mean when the population standard deviation is known. The formula is:
\[
\bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
\]
Where:
- \(\bar{x}\) is the sample mean.
- \(z\) is the z-value for the desired confidence level.
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
Given the information:
- Sample mean (\(\bar{x}\)) = 36 minutes
- Population standard deviation (\(\sigma\)) = 20 minutes
- Sample size (\(n\)) = 64
- Confidence level = 99%
### Step 1: Find the z-value for 99% confidence level
For a 99% confidence interval, the critical value \(z\) can be found using a Z-table or standard normal distribution calculator. The z-value corresponds to \(\frac{1 - 0.99}{2} = 0.005\) in each tail, which gives a z-value of approximately **2.576**.
### Step 2: Calculate the standard error (SE)
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{64}} = \frac{20}{8} = 2.5
\]
### Step 3: Calculate the margin of error (ME)
\[
ME = z \times SE = 2.576 \times 2.5 = 6.44
\]
### Step 4: Construct the confidence interval
\[
\text{Confidence Interval} = \bar{x} \pm ME = 36 \pm 6.44
\]
Thus, the confidence interval is:
\[
(36 - 6.44, 36 + 6.44) = (29.56, 42.44)
\]
So the 99% confidence interval for the actual average time it takes all motorists in Cape Town to drive to work is approximately (29.56 minutes, 42.44 minutes).
### Conclusion
The critical value for the 99% confidence interval is **2.576**.
\[
\bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
\]
Where:
- \(\bar{x}\) is the sample mean.
- \(z\) is the z-value for the desired confidence level.
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
Given the information:
- Sample mean (\(\bar{x}\)) = 36 minutes
- Population standard deviation (\(\sigma\)) = 20 minutes
- Sample size (\(n\)) = 64
- Confidence level = 99%
### Step 1: Find the z-value for 99% confidence level
For a 99% confidence interval, the critical value \(z\) can be found using a Z-table or standard normal distribution calculator. The z-value corresponds to \(\frac{1 - 0.99}{2} = 0.005\) in each tail, which gives a z-value of approximately **2.576**.
### Step 2: Calculate the standard error (SE)
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{64}} = \frac{20}{8} = 2.5
\]
### Step 3: Calculate the margin of error (ME)
\[
ME = z \times SE = 2.576 \times 2.5 = 6.44
\]
### Step 4: Construct the confidence interval
\[
\text{Confidence Interval} = \bar{x} \pm ME = 36 \pm 6.44
\]
Thus, the confidence interval is:
\[
(36 - 6.44, 36 + 6.44) = (29.56, 42.44)
\]
So the 99% confidence interval for the actual average time it takes all motorists in Cape Town to drive to work is approximately (29.56 minutes, 42.44 minutes).
### Conclusion
The critical value for the 99% confidence interval is **2.576**.