To determine the required sample size, we can use the formula for the sample size of a proportion:

$$ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) $$

where:
- $ n $ is the sample size,
- $ Z $ is the Z-value (the number of standard deviations from the mean) corresponding to the desired confidence level,
- $ p $ is the estimated proportion,
- $ E $ is the margin of error.

Given:
- The desired margin of error $ E $ is 0.03,
- The confidence level is 95%, which corresponds to a Z-value of approximately 1.96,
- The estimated proportion $ p $ from the previous study is 0.796.

Plugging these values into the formula:

$$ n = \left( \frac{(1.96)^2 \cdot 0.796 \cdot (1 - 0.796)}{(0.03)^2} \right) $$

First, calculate the numerator:

$$ (1.96)^2 = 3.8416 $$
$$ 0.796 \cdot (1 - 0.796) = 0.796 \cdot 0.204 = 0.162384 $$
$$ 3.8416 \cdot 0.162384 = 0.6238393344 $$

Next, calculate the denominator:

$$ (0.03)^2 = 0.0009 $$

Now, divide the numerator by the denominator:

$$ n = \frac{0.6238393344}{0.0009} \approx 693.155 $$

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

$$ n \approx 694 $$

Therefore, a sample size of 694 is required to estimate the proportion of first-year students who use certain types of social media within 0.03 of the true proportion with a 95% level of confidence.

Question

To determine the required sample size, we can use the formula for the sample size of a proportion:

$$ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) $$

where:
- $ n $ is the sample size,
- $ Z $ is the Z-value (the number of standard deviations from the mean) corresponding to the desired confidence level,
- $ p $ is the estimated proportion,
- $ E $ is the margin of error.

Given:
- The desired margin of error $ E $ is 0.03,
- The confidence level is 95%, which corresponds to a Z-value of approximately 1.96,
- The estimated proportion $ p $ from the previous study is 0.796.

Plugging these values into the formula:

$$ n = \left( \frac{(1.96)^2 \cdot 0.796 \cdot (1 - 0.796)}{(0.03)^2} \right) $$

First, calculate the numerator:

$$ (1.96)^2 = 3.8416 $$
$$ 0.796 \cdot (1 - 0.796) = 0.796 \cdot 0.204 = 0.162384 $$
$$ 3.8416 \cdot 0.162384 = 0.6238393344 $$

Next, calculate the denominator:

$$ (0.03)^2 = 0.0009 $$

Now, divide the numerator by the denominator:

$$ n = \frac{0.6238393344}{0.0009} \approx 693.155 $$

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

$$ n \approx 694 $$

Therefore, a sample size of 694 is required to estimate the proportion of first-year students who use certain types of social media within 0.03 of the true proportion with a 95% level of confidence.

Knowee AI · Accepted Answer