Question 1199774: A simple random sample of size n = 200 is obtained from a population whose size is N = 25,000 and whose population proportion with a specified characteristic is = = 0.65.
a) Describe the sampling distribution of 𝑝
b) What is the probability of obtaining x = 136 or fewer individuals with the characteristic? That is, what is P(𝑝 >= 0.68)?
c) What is the probability of obtaining x = 118 or fewer individuals with the characteristic? That is, what is P (p = 0.59)?
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a) Describe the sampling distribution of p**
* **Shape:** Since the sample size (n = 200) is large and the population size (N = 25,000) is much larger, the sampling distribution of the sample proportion (p̂) can be approximated by a normal distribution. This is due to the Central Limit Theorem.
* **Mean:** The mean of the sampling distribution of p̂ is equal to the population proportion (p):
* μp̂ = p = 0.65
* **Standard Deviation:** The standard deviation of the sampling distribution of p̂ is given by:
* σp̂ = √[p * (1 - p) / n]
* σp̂ = √[0.65 * (1 - 0.65) / 200]
* σp̂ = √[0.2275 / 200]
* σp̂ ≈ 0.0337
**Therefore, the sampling distribution of p̂ is approximately normal with mean 0.65 and standard deviation 0.0337.**
**b) Probability of obtaining x = 136 or fewer individuals with the characteristic (P(p̂ >= 0.68))**
1. **Calculate the z-score for p̂ = 0.68:**
* z = (p̂ - μp̂) / σp̂
* z = (0.68 - 0.65) / 0.0337
* z ≈ 0.89
2. **Find the probability:**
* P(p̂ >= 0.68) = P(Z >= 0.89)
* Using a standard normal table or calculator, find the area to the right of z = 0.89.
* P(Z >= 0.89) ≈ 0.1867
**Therefore, the probability of obtaining 136 or fewer individuals with the characteristic is approximately 0.1867.**
**c) Probability of obtaining x = 118 or fewer individuals with the characteristic (P(p̂ <= 0.59))**
1. **Calculate the z-score for p̂ = 0.59:**
* z = (0.59 - 0.65) / 0.0337
* z ≈ -1.78
2. **Find the probability:**
* P(p̂ <= 0.59) = P(Z <= -1.78)
* Using a standard normal table or calculator, find the area to the left of z = -1.78.
* P(Z <= -1.78) ≈ 0.0375
**Therefore, the probability of obtaining 118 or fewer individuals with the characteristic is approximately 0.0375.**
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