A survey of 480 high school students found that 37% had a pet. Find the margin of error. Round to the nearest percent. Use the margin of error to find an interval that is likely to contain the true population proportion.
* margin of error = +-5%
interval, 37% to 42%
* margin of error = +-5%
interval, 32% to 42%
* margin of error = +-7%
interval, 30% to 44%
* margin of error = +-7%
interval, 37% to 44%
To find the margin of error, we need to use the following formula:
Margin of Error = Critical Value * Standard Error
First, we need to find the critical value. The critical value depends on the desired level of confidence. Let's assume a 95% confidence level, which corresponds to a critical value of approximately 1.96 (for a large sample size).
Next, we need to calculate the standard error. The formula for the standard error is:
Standard Error = sqrt(p_hat * (1 - p_hat) / n)
where p_hat is the sample proportion (37% = 0.37) and n is the sample size (480).
Plugging in the values, we get:
Standard Error = sqrt(0.37 * (1 - 0.37) / 480)
= sqrt(0.2318 / 480)
≈ sqrt(0.000483)
Calculating this value, we find:
Standard Error ≈ 0.022
Now, we can calculate the margin of error:
Margin of Error = Critical Value * Standard Error
= 1.96 * 0.022
≈ 0.043
Rounding to the nearest percent, the margin of error is approximately 4%. This means that the true population proportion is likely to be within 4 percentage points of the sample proportion.
As for the interval, we take the sample proportion (37%) and add/subtract the margin of error (4%) to get an interval. This gives us the range of likely values for the true population proportion. From the provided options:
* margin of error = +-5%
interval, 37% to 42%
This is the correct interval that is likely to contain the true population proportion.