A simple random sample size of n = 150 is obtained from a population whose size is N = 20,000 and whose population proportion with a specified characteristic is p = 0.6.
What is the probability of obtaining x = 96 or more individuals with the characteristic
P(p hat)≥0.64
To calculate the probability of obtaining x = 96 or more individuals with the characteristic (P(p hat)≥0.64), we need to use the binomial distribution formula.
The formula for the binomial distribution is:
P(X = x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- n is the sample size (150 in this case)
- x is the number of individuals with the characteristic (96 or more in this case)
- p is the population proportion with the specified characteristic (0.6 in this case)
- (nCx) represents the number of combinations of n items taken x at a time, which can be calculated using the formula: (nCx) = n! / (x! * (n-x)!)
We want to calculate the probability of P(p hat)≥0.64, which means finding the probability of getting 96 or more individuals (96, 97, 98, ..., 150) out of the sample size of 150.
The probability of P(p hat)≥0.64 can be calculated by summing the individual probabilities of each value of x from 96 to 150.
P(P hat ≥ 0.64) = P(X = 96) + P(X = 97) + ... + P(X = 150)
To calculate each individual probability, we substitute the values of n, x, and p into the binomial distribution formula.
P(X = x) = (150Cx) * 0.6^x * (1-0.6)^(150-x)
We can use a statistical software program, calculator, or online binomial probability calculator to calculate the individual probabilities for each value of x from 96 to 150 and then sum them up to get the desired probability.
Alternatively, if you have access to statistical software like R or Python, you can use functions like dbinom() in R or scipy.stats.binom.pmf() in Python to calculate the individual probabilities and then sum them up to get the desired probability.
For example, in R, you can use the following code:
```
sum(dbinom(96:150, size = 150, prob = 0.6))
```
The result will give you the probability of obtaining x = 96 or more individuals with the specified characteristic.