Questions: For a confidence level of 90%, find the critical value. Round your answer to 2 decimal places. In a recent poll, 500 people were asked if they liked dogs, and 48% said they did. Find the margin of error of this poll, at the 99% confidence level. Give your answer to three decimals.

For a confidence level of 90%, find the critical value.

Round your answer to 2 decimal places.

In a recent poll, 500 people were asked if they liked dogs, and 48% said they did. Find the margin of error of this poll, at the 99% confidence level.

Give your answer to three decimals.

Transcript text: For a confidence level of $90 \%$, find the critical value. Round your answer to 2 decimal places. In a recent poll, 500 people were asked if they liked dogs, and $48 \%$ said they did. Find the margin of error of this poll, at the $99 \%$ confidence level. Give your answer to three decimals

Solution

Solution Steps

Step 1: Finding the Critical Value

To find the critical value for a confidence level of $90\%$, we use the formula for the Z critical value:

\[ Z = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right) \]

For a $90\%$ confidence level, the critical value is calculated as:

\[ Z = 1.64 \]

Step 2: Calculating the Margin of Error

In a recent poll, $500$ people were asked if they liked dogs, and $48\%$ said they did. We need to find the margin of error at a $99\%$ confidence level. The Z-score for a $99\%$ confidence level is:

\[ Z = 2.576 \]

The standard deviation for the proportion is calculated as:

\[ \sigma = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.48 \times (1 - 0.48)}{500}} \approx 0.0223 \]

The margin of error is then calculated using the formula:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} = \frac{2.576 \times 0.0223}{\sqrt{500}} \approx 0.003 \]