SOLUTION: A simple random sample of size n=49 is obtained from a population with mean = 80 and standard deviation = 14. a) Describe the sampling distribution of x̅. b) What is P(x̅

Algebra ->  Probability-and-statistics -> SOLUTION: A simple random sample of size n=49 is obtained from a population with mean = 80 and standard deviation = 14. a) Describe the sampling distribution of x̅. b) What is P(x̅       Log On


   

Question 1199773: A simple random sample of size n=49 is obtained from a population with mean = 80 and standard deviation = 14.
a) Describe the sampling distribution of x̅.
b) What is P(x̅ > 83)?
c) What is P(x̅ ≤ 75.8)?
d) What is P(78.3 < x̅ < 85.1)?
Answer by CPhill(1987) About Me  (Show Source):
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**a) Describe the sampling distribution of x̅**
* **Mean of the sampling distribution (μx̅):**
* μx̅ = μ = 80
* The mean of the sampling distribution of the sample means is equal to the population mean.
* **Standard deviation of the sampling distribution (standard error):**
* σx̅ = σ / √n = 14 / √49 = 14 / 7 = 2
* **Shape of the sampling distribution:**
* Since the sample size (n = 49) is sufficiently large (n ≥ 30), according to the Central Limit Theorem, the sampling distribution of the sample mean (x̅) will be approximately normally distributed, regardless of the shape of the original population distribution.
**Therefore:**
* The sampling distribution of x̅ is approximately normally distributed with:
* Mean (μx̅) = 80
* Standard deviation (σx̅) = 2
**b) P(x̅ > 83)**
1. **Standardize the value:**
* z = (x̅ - μx̅) / σx̅
* z = (83 - 80) / 2 = 1.5
2. **Find the probability using a standard normal distribution table or calculator:**
* P(x̅ > 83) = P(Z > 1.5)
* From the z-table, P(Z > 1.5) = 0.0668
**Therefore, P(x̅ > 83) = 0.0668**
**c) P(x̅ ≤ 75.8)**
1. **Standardize the value:**
* z = (75.8 - 80) / 2 = -2.1
2. **Find the probability using a standard normal distribution table or calculator:**
* P(x̅ ≤ 75.8) = P(Z ≤ -2.1)
* From the z-table, P(Z ≤ -2.1) = 0.0179
**Therefore, P(x̅ ≤ 75.8) = 0.0179**
**d) P(78.3 < x̅ < 85.1)**
1. **Standardize the values:**
* z1 = (78.3 - 80) / 2 = -0.85
* z2 = (85.1 - 80) / 2 = 2.55
2. **Find the probabilities using a standard normal distribution table or calculator:**
* P(x̅ < 85.1) = P(Z < 2.55)
* P(x̅ < 78.3) = P(Z < -0.85)
3. **Calculate the desired probability:**
* P(78.3 < x̅ < 85.1) = P(x̅ < 85.1) - P(x̅ < 78.3)
= P(Z < 2.55) - P(Z < -0.85)
≈ 0.9946 - 0.1977
≈ 0.7969
**Therefore, P(78.3 < x̅ < 85.1) = 0.7969**