To determine the required sample size, we can use the formula for the sample size of a population mean when the population standard deviation is known. The formula is:

$$ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 $$

where:
- $ n $ is the sample size,
- $ Z $ is the Z-value corresponding to the desired confidence level,
- $ \sigma $ is the population standard deviation,
- $ E $ is the margin of error.

Given:
- Population standard deviation ($ \sigma $) = 23 hours,
- Margin of error ($ E $) = 3 hours,
- Confidence level = 95%.

First, we need to find the Z-value for a 95% confidence level. The Z-value for a 95% confidence level is approximately 1.96.

Now, we can plug these values into the formula:

$$ n = \left( \frac{1.96 \cdot 23}{3} \right)^2 $$

Calculate the numerator:

$$ 1.96 \cdot 23 = 45.08 $$

Now, divide by the margin of error:

$$ \frac{45.08}{3} = 15.0267 $$

Finally, square the result:

$$ n = (15.0267)^2 \approx 225.8 $$

Since the sample size must be a whole number, we round up to the next whole number:

$$ n \approx 226 $$

Therefore, the required sample size is 226 students.

Question

To determine the required sample size, we can use the formula for the sample size of a population mean when the population standard deviation is known. The formula is:

$$ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 $$

where:
- $ n $ is the sample size,
- $ Z $ is the Z-value corresponding to the desired confidence level,
- $ \sigma $ is the population standard deviation,
- $ E $ is the margin of error.

Given:
- Population standard deviation ($ \sigma $) = 23 hours,
- Margin of error ($ E $) = 3 hours,
- Confidence level = 95%.

First, we need to find the Z-value for a 95% confidence level. The Z-value for a 95% confidence level is approximately 1.96.

Now, we can plug these values into the formula:

$$ n = \left( \frac{1.96 \cdot 23}{3} \right)^2 $$

Calculate the numerator:

$$ 1.96 \cdot 23 = 45.08 $$

Now, divide by the margin of error:

$$ \frac{45.08}{3} = 15.0267 $$

Finally, square the result:

$$ n = (15.0267)^2 \approx 225.8 $$

Since the sample size must be a whole number, we round up to the next whole number:

$$ n \approx 226 $$

Therefore, the required sample size is 226 students.

Knowee AI · Accepted Answer