To find the 99% confidence interval for the proportion of diamond cards drawn from the sample, we can follow these steps:
Step 1: Calculate the sample proportion (p̂)
The sample proportion \( p̂ \) of diamond cards is given by:
\[ p̂ = \frac{x}{n} \]
where \( x \) is the number of diamond cards (24) and \( n \) is the total number of cards drawn (100).
\[ p̂ = \frac{24}{100} = 0.24 \]
Step 2: Calculate the standard error (SE)
The standard error (SE) for the sample proportion is calculated using the formula:
\[ SE = \sqrt{\frac{p̂(1 - p̂)}{n}} \]
Substituting our values:
\[ SE = \sqrt{\frac{0.24(1 - 0.24)}{100}} = \sqrt{\frac{0.24 \times 0.76}{100}} = \sqrt{\frac{0.1824}{100}} = \sqrt{0.001824} \approx 0.0427 \]
Step 3: Determine the z-score for a 99% confidence level
For a 99% confidence interval, we find the z-score that corresponds to 0.5% in each tail of the normal distribution. The z-score for 99% confidence is approximately:
\[ z \approx 2.576 \]
Step 4: Calculate the margin of error (ME)
The margin of error (ME) can be calculated with:
\[ ME = z \times SE \]
Substituting the values:
\[ ME = 2.576 \times 0.0427 \approx 0.110 \]
Step 5: Calculate the confidence interval
Now we can calculate the confidence interval:
\[ \text{Lower bound} = p̂ - ME \ \text{Upper bound} = p̂ + ME \]
Calculating the bounds:
\[ \text{Lower bound} = 0.24 - 0.110 \approx 0.130 \ \text{Upper bound} = 0.24 + 0.110 \approx 0.350 \]
Final Rounded Confidence Interval
Rounding to the nearest thousandths:
- Lower bound: 0.130
- Upper bound: 0.350
Thus, the resulting 99% confidence interval for the proportion of diamond cards in the sample is:
- Lower bound: 0.130
- Upper bound: 0.350