Question 1186870: 1. The amount of syrup that people put on their pancakes is normally distributed with mean 63 mL and standard deviation 11 mL. Suppose that 50 randomly selected people are observed pouring syrup on their pancakes. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X-N( , )
b. What is the distribution of x¯? ¯x-N( , )
c. If a single randomly selected individual is observed, find the probability that this person consumes is between 61.8 mL and 64 mL?
d. For the group of 50 pancake eaters, find the probability that the average amount of syrup is between 61.8 mL and 64 mL?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's the solution:
**1. a. Distribution of X:**
X represents the amount of syrup used by a single person. We're given that it's normally distributed with a mean (μ) of 63 mL and a standard deviation (σ) of 11 mL. So:
X ~ N(63, 11)
**1. b. Distribution of x̄:**
x̄ represents the average amount of syrup used by a sample of 50 people. The distribution of the sample mean is also normal. The mean of x̄ is the same as the population mean (μ = 63). The standard deviation of x̄ (also called the standard error) is the population standard deviation (σ = 11) divided by the square root of the sample size (n = 50):
Standard Error = σ / √n = 11 / √50 ≈ 1.5556
So, x̄ ~ N(63, 1.5556)
**1. c. Probability for a single individual:**
We want to find P(61.8 < X < 64). First, convert these values to z-scores:
z₁ = (61.8 - 63) / 11 ≈ -0.1091
z₂ = (64 - 63) / 11 ≈ 0.0909
Now, look up these z-scores in a standard normal distribution table (or use a calculator) to find the corresponding probabilities:
P(z < -0.1091) ≈ 0.4565
P(z < 0.0909) ≈ 0.5362
P(61.8 < X < 64) = P(z < 0.0909) - P(z < -0.1091) = 0.5362 - 0.4565 ≈ 0.0797
**1. d. Probability for the average of 50 people:**
We want to find P(61.8 < x̄ < 64). Use the same z-score formula, but with the standard error of the mean:
z₁ = (61.8 - 63) / 1.5556 ≈ -0.7712
z₂ = (64 - 63) / 1.5556 ≈ 0.6422
P(z < -0.7712) ≈ 0.2200
P(z < 0.6422) ≈ 0.7396
P(61.8 < x̄ < 64) = P(z < 0.6422) - P(z < -0.7712) = 0.7396 - 0.2200 ≈ 0.5196
**Answers:**
a. X ~ N(63, 11)
b. x̄ ~ N(63, 1.5556)
c. 0.0797
d. 0.5196
|
|
|
