To solve this problem, we will calculate two things:

1. The point estimate for the proportion of distorted graphics.
2. The 95% confidence interval for the proportion of distorted graphics.

(1) Point Estimate

The point estimate for the proportion $\hat{p}$ is given by the formula:

$\hat{p} = \frac{x}{n}$

Where:

• $x = 142$ is the number of distorted graphics.
• $n = 453$ is the total number of graphics.

Now, calculate $\hat{p}$:

$\hat{p} = \frac{142}{453} \approx 0.3135$

So, the point estimate for the proportion of distorted graphics is 0.3135.

(2) 95% Confidence Interval

The formula for the confidence interval for a proportion is:

$\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $\hat{p}$ is the point estimate of the proportion (0.3135).
• $Z_{\alpha/2}$ is the critical value for the 95% confidence level (which is 1.96).
• $n$ is the sample size (453).

Now, calculate the margin of error:

$\text{Margin of error} = 1.96 \times \sqrt{\frac{0.3135(1 - 0.3135)}{453}}$

$= 1.96 \times \sqrt{\frac{0.3135 \times 0.6865}{453}}$

$= 1.96 \times \sqrt{\frac{0.2151}{453}}$

$= 1.96 \times \sqrt{0.0004748}$

$= 1.96 \times 0.02179 \approx 0.0427$

Now, calculate the confidence interval:

$\hat{p} \pm 0.0427$

$0.3135 - 0.0427 = 0.2708$ $0.3135 + 0.0427 = 0.3562$

So, the 95% confidence interval is approximately (0.2708, 0.3562).

Final Answer:

• The point estimate of the proportion of distorted graphics is 0.3135.
• The 95% confidence interval is (0.2708, 0.3562).

Would you like more details or have any further questions? Here are some related questions for deeper understanding:

1. How is the margin of error in a confidence interval influenced by sample size?
2. Why is the $Z_{\alpha/2}$ value 1.96 for a 95% confidence interval?
3. What does it mean if a confidence interval does not contain a certain value, like 0.30?
4. How does the confidence level (95% in this case) affect the width of the confidence interval?
5. Can the point estimate of a proportion be used to make predictions about a future sample?

Tip: A larger sample size will decrease the margin of error, leading to a more precise confidence interval!