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.

1 answer

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