If 80% of the applicants are able to pass a driver's proficiency road test, find the mean, variance, and standard deviation of the number of people who pass the test in a sample of 300 applicants.
Your values are the following:
p = .8, q = 1 - p = .2, and n = 300
Therefore:
mean = np = (300)(.8) = ?
variance = npq = (300)(.8)(.2)= ?
standard deviation = square root of the variance
I'll let you take it from here.
Answer
To find the mean, variance, and standard deviation of the number of people who pass the test in a sample of 300 applicants, we can use the formulas for a binomial distribution.
Given:
Probability of passing the test, P(X = pass) = 80% = 0.8
Number of trials, n = 300
Step 1: Mean (μ) calculation
The mean (μ) can be found using the formula:
μ = n * p
μ = 300 * 0.8
μ = 240
Step 2: Variance (σ^2) calculation
The variance (σ^2) can be found using the formula:
σ^2 = n * p * (1 - p)
σ^2 = 300 * 0.8 * (1 - 0.8)
σ^2 = 300 * 0.8 * 0.2
σ^2 = 48
Step 3: Standard Deviation (σ) calculation
The standard deviation (σ) can be found by taking the square root of the variance:
σ = √(σ^2)
σ = √48
σ ≈ 6.93
Therefore, the mean is 240, the variance is 48, and the standard deviation is approximately 6.93.
To find the mean, variance, and standard deviation of the number of people who pass the test in a sample of 300 applicants, we need to understand some basic concepts of probability and statistics.
Mean:
The mean, also known as the average, is calculated by summing up all the values in a dataset and dividing it by the number of observations. In this case, we know that 80% of the applicants are able to pass the test, which means 80% of 300, or 0.8 * 300 = 240 applicants pass the test. So, the mean is 240.
Variance:
Variance measures the spread of data from the mean. To calculate the variance, we need to find the difference between each data point and the mean, square those differences, sum them up, and divide by the number of observations. However, in this case, we are dealing with a binomial distribution because each applicant can either pass or fail the test.
For a binomial distribution, the variance (var) is calculated using the formula: var = np(1-p), where n is the number of trials (300 in this case), and p is the probability of success (80% or 0.8).
var = 300 * 0.8 * (1 - 0.8)
var = 300 * 0.8 * 0.2
var = 48
Standard Deviation:
The standard deviation is the square root of the variance and is used to measure the spread of data. So, the standard deviation (std) is the square root of the variance we calculated before.
std = √var
std = √48
std ≈ 6.93
Therefore, the mean is 240, the variance is 48, and the standard deviation is approximately 6.93.