Question 1171027: The local bakery bakes more than a thousand 1-pound loaves of bread daily, and the weights of these loaves varies. The mean weight is 1.9 lb. and 4 oz., or 975 grams. Assume the standard deviation of the weights is 26 grams and a sample of 35 loaves is to be randomly selected.
(b) Find the mean of this sampling distribution. (Give your answer correct to nearest whole number.)
(c) Find the standard error of this sampling distribution. (Give your answer correct to two decimal places.)
(d) What is the probability that this sample mean will be between 965 and 985? (Give your answer correct to four decimal places.)
(e) What is the probability that the sample mean will have a value less than 967? (Give your answer correct to four decimal places.)
(f) What is the probability that the sample mean will be within 3 grams of the mean? (Give your answer correct to four decimal places.)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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 (μx̄) is equal to the population mean (μ).
* μx̄ = μ = 975
Therefore, the mean of the sampling distribution is 975.
**(c) Find the Standard Error of the Sampling Distribution**
The standard error (σx̄) is the standard deviation of the sampling distribution of the sample mean. It's calculated as:
* σx̄ = σ / √n
* σx̄ = 26 / √35
* σx̄ ≈ 26 / 5.9161
* σx̄ ≈ 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
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