Question
Given the population standard deviation is known as R500. A random sample of 50 individuals resulted in a mean of R10000. What is the width of the 90% confidence interval?
R232.64
R364.30
R465.23
R728.60
R232.64
R364.30
R465.23
R728.60
Answers
Answered by
GPT-4o mini
To calculate the width of the 90% confidence interval for the sample mean, we can use the formula for the confidence interval when the population standard deviation is known. The formula is:
\[
\text{Confidence Interval} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)
\]
Where:
- \(\bar{x}\) is the sample mean,
- \(Z\) is the z-value corresponding to the desired confidence level,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
1. **Determine the z-value for 90% confidence level**:
For a 90% confidence interval, the z-value (critical value) can be found in standard z-tables or using statistical calculators. The z-value corresponding to a 90% confidence level is approximately 1.645.
2. **Plug the values into the formula**:
- \(\bar{x} = R 10000\)
- \(\sigma = R 500\)
- \(n = 50\)
First, calculate the standard error (SE):
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{500}{\sqrt{50}} \approx \frac{500}{7.071} \approx 70.71
\]
3. **Calculate the margin of error (ME)**:
\[
ME = Z \cdot SE = 1.645 \times 70.71 \approx 116.52
\]
4. **Calculate the width of the confidence interval**:
The width of the confidence interval is twice the margin of error:
\[
\text{Width} = 2 \times ME = 2 \times 116.52 \approx 233.04
\]
Now let’s see if there are discrepancies since the options provided do not include 233.04. This indicates that it could be useful to double-check calculations.
**Rechecking Margin of Error Calculation**:
1. **Recalculating Standard Error**:
\[
SE = \frac{500}{\sqrt{50}} \approx 70.71
\]
2. **Recalculating Margin of Error**:
\[
ME = Z \cdot SE = 1.645 \times 70.71 \approx 116.52
\]
3. **Recalculating Width**:
\[
\text{Width} = 2 \times 116.52 = 233.04
\]
Now checking recommended choices again:
Given the values:
- R232.64
- R364.30
- R465.23
- R728.60
The value R232.64 is the closest to our calculated confidence width.
Therefore, the closest answer choice for the width of the 90% confidence interval would be:
**R232.64**
\[
\text{Confidence Interval} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)
\]
Where:
- \(\bar{x}\) is the sample mean,
- \(Z\) is the z-value corresponding to the desired confidence level,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
1. **Determine the z-value for 90% confidence level**:
For a 90% confidence interval, the z-value (critical value) can be found in standard z-tables or using statistical calculators. The z-value corresponding to a 90% confidence level is approximately 1.645.
2. **Plug the values into the formula**:
- \(\bar{x} = R 10000\)
- \(\sigma = R 500\)
- \(n = 50\)
First, calculate the standard error (SE):
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{500}{\sqrt{50}} \approx \frac{500}{7.071} \approx 70.71
\]
3. **Calculate the margin of error (ME)**:
\[
ME = Z \cdot SE = 1.645 \times 70.71 \approx 116.52
\]
4. **Calculate the width of the confidence interval**:
The width of the confidence interval is twice the margin of error:
\[
\text{Width} = 2 \times ME = 2 \times 116.52 \approx 233.04
\]
Now let’s see if there are discrepancies since the options provided do not include 233.04. This indicates that it could be useful to double-check calculations.
**Rechecking Margin of Error Calculation**:
1. **Recalculating Standard Error**:
\[
SE = \frac{500}{\sqrt{50}} \approx 70.71
\]
2. **Recalculating Margin of Error**:
\[
ME = Z \cdot SE = 1.645 \times 70.71 \approx 116.52
\]
3. **Recalculating Width**:
\[
\text{Width} = 2 \times 116.52 = 233.04
\]
Now checking recommended choices again:
Given the values:
- R232.64
- R364.30
- R465.23
- R728.60
The value R232.64 is the closest to our calculated confidence width.
Therefore, the closest answer choice for the width of the 90% confidence interval would be:
**R232.64**