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We know that proportion of diamond cards in a regular deck of cards is 13/52=0.25. Assume for a moment that we do not know it and we wish to use statistical methods to find a confidence interval for proportion of red cards in a deck. We will use 10 regular decks of card and select random sample of 100 cards. We can use the methods used in this class for finding confidence interval because all assumptions are satisfied. Namely,

1) the sample is random

2) the values in the sample are independent

3) np̂ ≥
and n(1-p̂)≥

where n- sample size; p̂ -sample proportion

1 answer

To find a confidence interval for the proportion of red cards in a deck of cards based on a random sample of 100 cards from 10 regular decks, we need to follow a series of steps. We will assume that the observations and assumptions you mentioned hold, particularly about the independence and randomness of sample selection.

Definitions

  1. Sample Size (n): 100 cards
  2. Sample Proportion (\(\hat{p}\)): The proportion of red cards found in the sample.
  3. Confidence Level: Usually, confidence levels are set at 90%, 95%, or 99%. For this example, we can use a 95% confidence level.

Steps to Create the Confidence Interval

  1. Collect Data: Randomly select 100 cards and count the number of red cards (hearts and diamonds). Let’s denote the count of red cards as \(X\).

  2. Calculate Sample Proportion: \[ \hat{p} = \frac{X}{n} \] where \(n = 100\).

  3. Check Conditions for Normal Approximation: Ensure the conditions \(n \hat{p} \geq 5\) and \(n (1 - \hat{p}) \geq 5\) are satisfied. This ensures the sampling distribution of the proportion can be approximated by a normal distribution.

  4. Calculate the Standard Error (SE): \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

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

  6. Calculate the Confidence Interval: The confidence interval can be computed as: \[ \hat{p} \pm Z \cdot SE \] So, the lower and upper bounds of the confidence interval will be: \[ \text{Lower Bound} = \hat{p} - Z \cdot SE \] \[ \text{Upper Bound} = \hat{p} + Z \cdot SE \]

Example Calculation

Assuming you sampled 100 cards and found 48 red cards:

  1. Calculate \(\hat{p}\): \[ \hat{p} = \frac{48}{100} = 0.48 \]

  2. Check the conditions: \[ n \hat{p} = 100 \cdot 0.48 = 48 \quad (\geq 5) \] \[ n (1 - \hat{p}) = 100 \cdot 0.52 = 52 \quad (\geq 5) \]

  3. Calculate SE: \[ SE = \sqrt{\frac{0.48 \cdot 0.52}{100}} \approx \sqrt{0.002496} \approx 0.04996 \]

  4. Calculate Confidence Interval: \[ \text{Lower Bound} = 0.48 - 1.96 \cdot 0.04996 \approx 0.48 - 0.0978 \approx 0.3822 \] \[ \text{Upper Bound} = 0.48 + 1.96 \cdot 0.04996 \approx 0.48 + 0.0978 \approx 0.5778 \]

Thus, the 95% confidence interval for the proportion of red cards in the deck based on the sample would be approximately \( (0.382, 0.578) \).

Conclusion

Using these steps, you can find a confidence interval for the proportion of red cards with the data from your random sample. Make sure to replace the example values with your actual sample data to get your specific confidence interval.