Question 1171028
Let's solve this problem step-by-step.

**Given Information:**

* Population mean (μ) = 55
* Population standard deviation (σ) = 15
* Sample size (n) = 43

**(b) Find the Mean of the Sampling Distribution**

The mean of the sampling distribution of the sample mean (μ<sub>x̄</sub>) is equal to the population mean (μ).

* μ<sub>x̄</sub> = μ = 55

Therefore, the mean of the sampling distribution is 55.

**(c) Find the Standard Error of the Sampling Distribution**

The standard error (σ<sub>x̄</sub>) is the standard deviation of the sampling distribution of the sample mean. It's calculated as:

* σ<sub>x̄</sub> = σ / √n
* σ<sub>x̄</sub> = 15 / √43
* σ<sub>x̄</sub> ≈ 15 / 6.5574
* σ<sub>x̄</sub> ≈ 2.2875

Rounded to two decimal places, the standard error is 2.29.

**(d) Probability that the Sample Mean is Between 43 and 51**

We need to find P(43 < x̄ < 51). First, we need to convert the sample means to z-scores:

* z₁ = (43 - 55) / 2.29 = -12 / 2.29 ≈ -5.24
* z₂ = (51 - 55) / 2.29 = -4 / 2.29 ≈ -1.75

Now, we need to find P(-5.24 < Z < -1.75).

* P(Z < -1.75) ≈ 0.0401
* P(Z < -5.24) ≈ 0 (very close to 0)

Therefore, P(-5.24 < Z < -1.75) = P(Z < -1.75) - P(Z < -5.24) ≈ 0.0401 - 0 ≈ 0.0401

**(e) Probability that the Sample Mean is Greater Than 51**

We need to find P(x̄ > 51). We already calculated the z-score for 51: z = -1.75.

* P(Z > -1.75) = 1 - P(Z < -1.75)
* P(Z > -1.75) ≈ 1 - 0.0401 = 0.9599

**(f) Probability that the Sample Mean is Within 3 Units of the Mean**

We need to find P(55 - 3 < x̄ < 55 + 3), which is P(52 < x̄ < 58).

* z₁ = (52 - 55) / 2.29 = -3 / 2.29 ≈ -1.31
* z₂ = (58 - 55) / 2.29 = 3 / 2.29 ≈ 1.31

We need to find P(-1.31 < Z < 1.31).

* P(Z < 1.31) ≈ 0.9049
* P(Z < -1.31) ≈ 0.0951

Therefore, P(-1.31 < Z < 1.31) = P(Z < 1.31) - P(Z < -1.31) ≈ 0.9049 - 0.0951 = 0.8098

**Answers:**

(b) 55
(c) 2.29
(d) 0.0401
(e) 0.9599
(f) 0.8098