Random samples of size n = 90 were selected from a binomial population with p = 0.4. Use the normal distribution to approximate the following probabilities. (Round your answers to four decimal places.)
(a)
P(p̂ ≤ 0.49) =
(b)
P(0.37 ≤ p̂ ≤ 0.49) =
Thanks.
423
To solve these probability problems using the normal distribution approximation, we need to use the mean and standard deviation of the binomial distribution to standardize the values and then use the standard normal distribution table.
Here's how you can calculate the probabilities:
Step 1: Calculate the mean and standard deviation of the binomial distribution.
For a binomial distribution, the mean (μ) is given by n * p, and the standard deviation (σ) is given by the square root of (n * p * (1 - p)). In this case, n = 90 and p = 0.4, so:
μ = 90 * 0.4 = 36
σ = √(90 * 0.4 * (1 - 0.4)) = √(54) ≈ 7.3485 (rounded to four decimal places).
Step 2: Standardize the values.
To standardize a value x using the mean (μ) and standard deviation (σ), use the formula:
z = (x - μ) / σ.
(a) For the probability P(p̂ ≤ 0.49):
Standardize the value 0.49:
z = (0.49 - 36) / 7.3485 ≈ -4.5818 (rounded to four decimal places).
(b) For the probability P(0.37 ≤ p̂ ≤ 0.49):
Standardize the lower value, 0.37:
z1 = (0.37 - 36) / 7.3485 ≈ -4.6321 (rounded to four decimal places).
Standardize the upper value, 0.49:
z2 = (0.49 - 36) / 7.3485 ≈ -4.5818 (rounded to four decimal places).
Step 3: Use the standard normal distribution table.
Look up the probabilities corresponding to the standardized values from the standard normal distribution table. For each probability, use the cumulative distribution function (CDF) to find the area under the curve up to that standardized value.
(a) P(p̂ ≤ 0.49):
Using the standard normal distribution table, the probability corresponding to z ≈ -4.5818 is approximately 0.0000.
(b) P(0.37 ≤ p̂ ≤ 0.49):
Using the standard normal distribution table, the probability corresponding to z1 ≈ -4.6321 is approximately 0.0000, and the probability corresponding to z2 ≈ -4.5818 is approximately 0.0000.
Remember to round your answers to four decimal places, as specified.
So the answers to the given problems are:
(a) P(p̂ ≤ 0.49) ≈ 0.0000
(b) P(0.37 ≤ p̂ ≤ 0.49) ≈ 0.0000
Please note that the probabilities obtained using the normal distribution approximation are approximations and may not be exact. The approximation becomes more accurate as the sample size increases.