From a sample of 100 Bloemfontein car commuters, the sample mean time to commute to work daily was found to be 35.8 minutes. It is known that the population standard deviation is 20 minutes.

Question

Find the margin of error with a 95% confidence level.

Answer 1

To calculate the margin of error (ME) at a 95% confidence level, we can use the formula:

\[ ME = z \times \frac{\sigma}{\sqrt{n}} \]

where:

• \( z \) 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 a 95% confidence level: For a 95% confidence level, the z-value (which you can find in the z-table) is approximately 1.96.

2. Identify the given values:

   • Population standard deviation (\( \sigma \)) = 20 minutes
   • Sample size (\( n \)) = 100

3. Calculate the standard error (SE):

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2 \]

4. Calculate the margin of error (ME):

\[ ME = z \times SE = 1.96 \times 2 = 3.92 \]

So, the margin of error with a 95% confidence level is 3.92 minutes.