Question

The population will consider of 10 regular decks of cards well shuffled..

As described, we randomly draw 100 cards from the population and record a sample (D - diamond card, ND - not diamond card}. In the sample we have 24 diamond cards and 76 of the other suite.Find the 99% confidence interval for your sample.

Lower bound of the interval is

Upper bound of the interval is

Please round your answers to nearest thousandths.

Answer 1

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**