Use the given values of n and p to find the minimum usual value μ - 2σ and the maximum usual value μ + 2σ. Round your answer to the nearest hundredth unless otherwise noted.
n = 1056, p = 0.80
To find the minimum and maximum usual values, we first find the mean (μ) and standard deviation (σ) of the given data.
Mean (μ) = n * p,
where n = 1056 (number of trials) and p = 0.80 (probability of success).
μ = 1056 * 0.80 = 844.8
Standard deviation (σ) is given by the formula: σ = √(n * p * (1 - p))
σ = √(1056 * 0.80 * (1 - 0.80))
σ = √(844.8 * 0.20)
σ = √(168.96)
σ ≈ 12.997
Now that we have the mean and standard deviation, we can find the minimum and maximum usual values:
Minimum usual value (μ - 2σ) = 844.8 - 2 * 12.997
Minimum usual value ≈ 844.8 - 25.994
Minimum usual value ≈ 818.81 (rounded to the nearest hundredth)
Maximum usual value (μ + 2σ) = 844.8 + 2 * 12.997
Maximum usual value ≈ 844.8 + 25.994
Maximum usual value ≈ 870.79 (rounded to the nearest hundredth)
So, the minimum usual value is around 818.81 and the maximum usual value is around 870.79.
To find the minimum usual value μ - 2σ and the maximum usual value μ + 2σ, we need to calculate the mean (μ) and standard deviation (σ) first.
The formula to calculate the mean (μ) is given by:
μ = n * p
Substituting the given values, we have:
μ = 1056 * 0.80
μ = 844.8
The formula to calculate the standard deviation (σ) is given by:
σ = √(n * p * (1 - p))
Substituting the given values, we have:
σ = √(1056 * 0.80 * (1 - 0.80))
σ = √(844.8 * 0.20)
σ = √(168.96)
σ = 12.99
Now we can calculate the minimum usual value (μ - 2σ):
Minimum Usual Value = μ - 2σ
Minimum Usual Value = 844.8 - 2 * 12.99
Minimum Usual Value = 844.8 - 25.98
Minimum Usual Value ≈ 818.82
And the maximum usual value (μ + 2σ):
Maximum Usual Value = μ + 2σ
Maximum Usual Value = 844.8 + 2 * 12.99
Maximum Usual Value = 844.8 + 25.98
Maximum Usual Value ≈ 870.78
Therefore, the minimum usual value μ - 2σ is approximately 818.82, and the maximum usual value μ + 2σ is approximately 870.78.
To find the minimum usual value (μ - 2σ) and the maximum usual value (μ + 2σ), we'll need to use the formula of the confidence interval for a proportion.
The formula for the confidence interval is given by:
CI = p ± z * √(p * (1 - p) / n)
Where:
- CI represents the confidence interval,
- p is the proportion or percentage,
- z is the z-score corresponding to the desired confidence level,
- √(p * (1 - p) / n) represents the standard error.
Since we want μ - 2σ and μ + 2σ, we'll use a confidence level of 95%, which corresponds to a z-score of 1.96.
Let's calculate μ - 2σ first:
CI = p - z * √(p * (1 - p) / n)
= 0.8 - 1.96 * √(0.8 * (1 - 0.8) / 1056)
Calculate the expression inside the square root:
= 0.8 - 1.96 * √(0.16 / 1056)
= 0.8 - 1.96 * √(0.00015)
Evaluate the square root:
= 0.8 - 1.96 * 0.01225
≈ 0.8 - 0.02402
≈ 0.77598
Rounding to the nearest hundredth, μ - 2σ ≈ 0.78.
Now, let's calculate μ + 2σ:
CI = p + z * √(p * (1 - p) / n)
= 0.8 + 1.96 * √(0.8 * (1 - 0.8) / 1056)
Calculate the expression inside the square root:
= 0.8 + 1.96 * √(0.16 / 1056)
= 0.8 + 1.96 * √(0.00015)
Evaluate the square root:
= 0.8 + 1.96 * 0.01225
≈ 0.8 + 0.02402
≈ 0.82402
Rounding to the nearest hundredth, μ + 2σ ≈ 0.82.
Therefore, the minimum usual value (μ - 2σ) is approximately 0.78 and the maximum usual value (μ + 2σ) is approximately 0.82.