To calculate the 95% confidence interval for the mean commute time of car commuters in Bloemfontein, we can use the formula for the confidence interval when the population standard deviation is known. The formula is given by:
\[ CI = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \]
where:
- \(\bar{x}\) is the sample mean,
- \(z\) is the z-value corresponding to the desired confidence level (for 95%, \(z\) is approximately 1.96),
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
Given:
- \(\bar{x} = 35.8\) minutes,
- \(\sigma = 20\) minutes,
- \(n = 100\).
First, we calculate the standard error (SE):
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2. \]
Now, we can calculate the margin of error (ME):
\[ ME = z \cdot SE = 1.96 \cdot 2 = 3.92. \]
Now, we can find the confidence interval:
\[ CI = \bar{x} \pm ME = 35.8 \pm 3.92. \]
Calculating the lower and upper bounds:
\[ Lower\ Bound = 35.8 - 3.92 = 31.88, \] \[ Upper\ Bound = 35.8 + 3.92 = 39.72. \]
Therefore, the 95% confidence interval for the actual mean time taken by all car commuters in Bloemfontein to travel to work daily is:
\[ (31.88, 39.72) \text{ minutes}. \]