Question
9 sample size
199
169
385
329
269
149
135
249
349
Construct a 99.5% confidence interval
199
169
385
329
269
149
135
249
349
Construct a 99.5% confidence interval
Answers
GPT-4o mini
To construct a 99.5% confidence interval for the mean of the given sample, you can follow these steps:
1. **Calculate the sample mean (\(\bar{x}\))**.
2. **Calculate the sample standard deviation (s)**.
3. **Determine the appropriate z-score for a 99.5% confidence level**.
4. **Use the formula for the confidence interval**.
### Step 1: Calculate the Sample Mean
The sample mean \(\bar{x}\) is calculated as follows:
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{199 + 169 + 385 + 329 + 269 + 149 + 135 + 249 + 349}{9}
\]
Calculating the sum:
\[
199 + 169 + 385 + 329 + 269 + 149 + 135 + 249 + 349 = 1839
\]
Now divide by the sample size:
\[
\bar{x} = \frac{1839}{9} \approx 204.33
\]
### Step 2: Calculate the Sample Standard Deviation (s)
The sample standard deviation \(s\) can be calculated with the formula:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]
Calculating each term \((x_i - \bar{x})^2\):
\[
\begin{align*}
(199 - 204.33)^2 & \approx 28.51 \\
(169 - 204.33)^2 & \approx 1262.51 \\
(385 - 204.33)^2 & \approx 32757.71 \\
(329 - 204.33)^2 & \approx 15562.19 \\
(269 - 204.33)^2 & \approx 4261.51 \\
(149 - 204.33)^2 & \approx 3031.71 \\
(135 - 204.33)^2 & \approx 4802.31 \\
(249 - 204.33)^2 & \approx 1997.21 \\
(349 - 204.33)^2 & \approx 20756.51 \\
\end{align*}
\]
Now, summing these squared differences:
\[
28.51 + 1262.51 + 32757.71 + 15562.19 + 4261.51 + 3031.71 + 4802.31 + 1997.21 + 20756.51 \approx 68760.67
\]
Now calculate \(s\):
\[
s = \sqrt{\frac{68760.67}{9-1}} = \sqrt{\frac{68760.67}{8}} \approx \sqrt{8595.08} \approx 92.69
\]
### Step 3: Determine the Z-Score for 99.5% Confidence Level
For a \(99.5\%\) confidence interval, the significance level \(\alpha = 0.005\). Thus, for a two-tailed test:
\[
\alpha/2 = 0.0025
\]
Using the z-table or calculator, the z-score corresponding to \(0.0025\) in each tail is approximately \(2.807\).
### Step 4: Calculate the Confidence Interval
The formula for the confidence interval is:
\[
\bar{x} \pm z \frac{s}{\sqrt{n}}
\]
Plugging in the values:
\[
\text{Margin of Error} = z \cdot \frac{s}{\sqrt{n}} = 2.807 \cdot \frac{92.69}{\sqrt{9}} \approx 2.807 \cdot 30.90 \approx 86.73
\]
Finally, construct the confidence interval:
\[
\text{Lower Limit} = \bar{x} - \text{Margin of Error} \approx 204.33 - 86.73 \approx 117.60
\]
\[
\text{Upper Limit} = \bar{x} + \text{Margin of Error} \approx 204.33 + 86.73 \approx 291.06
\]
### Conclusion
The \(99.5\%\) confidence interval for the mean is approximately:
\[
(117.60, 291.06)
\]
1. **Calculate the sample mean (\(\bar{x}\))**.
2. **Calculate the sample standard deviation (s)**.
3. **Determine the appropriate z-score for a 99.5% confidence level**.
4. **Use the formula for the confidence interval**.
### Step 1: Calculate the Sample Mean
The sample mean \(\bar{x}\) is calculated as follows:
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{199 + 169 + 385 + 329 + 269 + 149 + 135 + 249 + 349}{9}
\]
Calculating the sum:
\[
199 + 169 + 385 + 329 + 269 + 149 + 135 + 249 + 349 = 1839
\]
Now divide by the sample size:
\[
\bar{x} = \frac{1839}{9} \approx 204.33
\]
### Step 2: Calculate the Sample Standard Deviation (s)
The sample standard deviation \(s\) can be calculated with the formula:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]
Calculating each term \((x_i - \bar{x})^2\):
\[
\begin{align*}
(199 - 204.33)^2 & \approx 28.51 \\
(169 - 204.33)^2 & \approx 1262.51 \\
(385 - 204.33)^2 & \approx 32757.71 \\
(329 - 204.33)^2 & \approx 15562.19 \\
(269 - 204.33)^2 & \approx 4261.51 \\
(149 - 204.33)^2 & \approx 3031.71 \\
(135 - 204.33)^2 & \approx 4802.31 \\
(249 - 204.33)^2 & \approx 1997.21 \\
(349 - 204.33)^2 & \approx 20756.51 \\
\end{align*}
\]
Now, summing these squared differences:
\[
28.51 + 1262.51 + 32757.71 + 15562.19 + 4261.51 + 3031.71 + 4802.31 + 1997.21 + 20756.51 \approx 68760.67
\]
Now calculate \(s\):
\[
s = \sqrt{\frac{68760.67}{9-1}} = \sqrt{\frac{68760.67}{8}} \approx \sqrt{8595.08} \approx 92.69
\]
### Step 3: Determine the Z-Score for 99.5% Confidence Level
For a \(99.5\%\) confidence interval, the significance level \(\alpha = 0.005\). Thus, for a two-tailed test:
\[
\alpha/2 = 0.0025
\]
Using the z-table or calculator, the z-score corresponding to \(0.0025\) in each tail is approximately \(2.807\).
### Step 4: Calculate the Confidence Interval
The formula for the confidence interval is:
\[
\bar{x} \pm z \frac{s}{\sqrt{n}}
\]
Plugging in the values:
\[
\text{Margin of Error} = z \cdot \frac{s}{\sqrt{n}} = 2.807 \cdot \frac{92.69}{\sqrt{9}} \approx 2.807 \cdot 30.90 \approx 86.73
\]
Finally, construct the confidence interval:
\[
\text{Lower Limit} = \bar{x} - \text{Margin of Error} \approx 204.33 - 86.73 \approx 117.60
\]
\[
\text{Upper Limit} = \bar{x} + \text{Margin of Error} \approx 204.33 + 86.73 \approx 291.06
\]
### Conclusion
The \(99.5\%\) confidence interval for the mean is approximately:
\[
(117.60, 291.06)
\]