Question

Katie is an eccentric statistics professor with an extensive comic book collection. She has a strange theory that a certain comic book artist does not like drawing feet. To investigate this, she looked at 75 randomly chosen panels by the artist and checked whether feet were visible in the panel. Visible feet Panels yes 19 no 56 Find a 95% confidence interval for p, the proportion of panels by the artist that do not include feet.

Answer 1

The 95% confidence interval for the proportion p of panels without visible feet is approximately 0.509 to 0.707.

To calculate a confidence interval for the proportion p of panels without visible feet, you can use the formula for the confidence interval for a population proportion:

`\[ \text{Confidence Interval} = \hat{p} \pm Z * \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]`

where:

-
`\( \hat{p} \)` is the sample proportion,

- Z is the Z-score corresponding to the desired confidence level,

- n is the sample size.

Given:

-
`\( \hat{p} = \frac{\text{Number of panels without feet}}{\text{Total number of panels}} = (56)/(75) \),`

- n = 75,

- For a 95% confidence interval, the Z-score is approximately 1.96.

Now, plug these values into the formula:

`\[ \text{Confidence Interval} = (56)/(75) \pm 1.96 * \sqrt{((56)/(75) * (19)/(75))/(75)} \]`

Calculate the values:

`\[ \text{Confidence Interval} = (56)/(75) \pm 1.96 * \sqrt{(56 * 19)/(75^3)} \]`

`\[ \text{Confidence Interval} = (56)/(75) \pm 1.96 * \sqrt{(1064)/(421875)} \]`

`\[ \text{Confidence Interval} = (56)/(75) \pm 1.96 * (√(1064))/(687.5) \]`

`\[ \text{Confidence Interval} \approx (56)/(75) \pm 0.148 \]`

Now, compute the interval:

`\[ \text{Lower bound} \approx (56)/(75) - 0.148 \]`

`\[ \text{Upper bound} \approx (56)/(75) + 0.148 \]`

So, the 95% confidence interval for the proportion p of panels without visible feet is approximately 0.509 to 0.707.