In a certain city of several million people, of the adults are unemployed. If a random sample of adults in this city is selected, approximate the probability that at least in the sample are unemployed. Use the normal approximation to the binomial with a correction for continuity.
Round your answer to at least three decimal places. Do not round any intermediate steps.
Data missing.
To approximate the probability, we need to use the normal approximation to the binomial distribution. The steps to approximate this probability are as follows:
1. Identify the parameters:
- Sample size: This is not provided in the question, so we'll assume it to be large enough for the normal approximation.
- Probability of success (unemployed adult): This is not provided in the question either. We'll refer to it as "p."
2. Calculate the mean (μ) and standard deviation (σ) of the distribution:
- Mean (μ) = n * p
- Standard deviation (σ) = √(n * p * (1 - p))
3. Apply the continuity correction:
- When using the normal approximation to the binomial distribution, we need to adjust the boundaries of the probability. Since the question asks for "at least k unemployed," we subtract 0.5 from k.
4. Standardize the boundaries:
- Calculate the z-scores for both the lower boundary (k - 0.5) and the upper boundary (infinity).
5. Use the standard normal distribution to find the probabilities:
- Subtract the cumulative probability corresponding to the lower boundary z-score from 1 to get the probability of "at least k unemployed."
Let's label the variables:
n = sample size
p = probability of an unemployed adult in the city
k = minimum number of unemployed adults in the sample
Since the values of n, p, and k are not given, let's assume some values for illustration purposes:
n = 1000
p = 0.1 (10% of adults are unemployed)
k = 100 (minimum number of unemployed adults in the sample)
Now, we can proceed with the calculations:
1. Calculate the mean and standard deviation:
- Mean (μ) = n * p = 1000 * 0.1 = 100
- Standard deviation (σ) = √(n * p * (1 - p)) = √(1000 * 0.1 * 0.9) ≈ 9.4868
2. Apply the continuity correction:
- Lower boundary = k - 0.5 = 100 - 0.5 = 99.5
3. Standardize the boundaries:
- For the lower boundary: z-score = (x - μ) / σ = (99.5 - 100) / 9.4868 = -0.0529 (rounded to four decimal places)
- For the upper boundary: Since the upper boundary is infinity, we can ignore it for this calculation.
4. Use the standard normal distribution to find the probabilities:
- Subtract the cumulative probability corresponding to the lower boundary z-score from 1:
P(x ≥ 100) = 1 - P(z ≤ -0.0529)
5. Use a standard normal distribution table, calculator, or software to find the cumulative probability corresponding to the z-score of -0.0529. The result is the probability of "at least 100 unemployed adults in the sample."
Remember to round the final answer to at least three decimal places as requested.