Question 1171027
Let's solve this problem step-by-step.

**Given Information:**

* Population mean (μ) = 975 grams
* Population standard deviation (σ) = 26 grams
* Sample size (n) = 35

**(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 = 975

Therefore, the mean of the sampling distribution is 975.

**(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> = 26 / √35
* σ<sub>x̄</sub> ≈ 26 / 5.9161
* σ<sub>x̄</sub> ≈ 4.3946

Rounded to two decimal places, the standard error is 4.39.

**(d) Probability that the Sample Mean is Between 965 and 985**

We need to find P(965 < x̄ < 985). First, we need to convert the sample means to z-scores:

* z₁ = (965 - 975) / 4.39 = -10 / 4.39 ≈ -2.28
* z₂ = (985 - 975) / 4.39 = 10 / 4.39 ≈ 2.28

Now, we need to find P(-2.28 < Z < 2.28).

* P(Z < 2.28) ≈ 0.9887
* P(Z < -2.28) ≈ 0.0113

Therefore, P(-2.28 < Z < 2.28) = P(Z < 2.28) - P(Z < -2.28) ≈ 0.9887 - 0.0113 = 0.9774

**(e) Probability that the Sample Mean is Less Than 967**

We need to find P(x̄ < 967). First, we need to convert 967 to a z-score:

* z = (967 - 975) / 4.39 = -8 / 4.39 ≈ -1.82

Now, we need to find P(Z < -1.82).

* P(Z < -1.82) ≈ 0.0344

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

We need to find P(975 - 3 < x̄ < 975 + 3), which is P(972 < x̄ < 978).

* z₁ = (972 - 975) / 4.39 = -3 / 4.39 ≈ -0.68
* z₂ = (978 - 975) / 4.39 = 3 / 4.39 ≈ 0.68

We need to find P(-0.68 < Z < 0.68).

* P(Z < 0.68) ≈ 0.7517
* P(Z < -0.68) ≈ 0.2483

Therefore, P(-0.68 < Z < 0.68) = P(Z < 0.68) - P(Z < -0.68) ≈ 0.7517 - 0.2483 = 0.5034

**Answers:**

(b) 975
(c) 4.39
(d) 0.9774
(e) 0.0344
(f) 0.5034