Question 1199773
**a) Describe the sampling distribution of x̅**

* **Mean of the sampling distribution (μ<sub>x̅</sub>):** 
    * μ<sub>x̅</sub> = 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):** 
    * σ<sub>x̅</sub> = 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 (μ<sub>x̅</sub>) = 80
    * Standard deviation (σ<sub>x̅</sub>) = 2 

**b) P(x̅ > 83)**

1. **Standardize the value:**
    * z = (x̅ - μ<sub>x̅</sub>) / σ<sub>x̅</sub> 
    * 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**