Question 1186871: 1. Suppose the age that children learn to walk is normally distributed with mean 12 months and standard deviation 1.5 month. 13 randomly selected people were asked what age they learned to walk. Round all answers to 4 decimal places where possible.
b. What is the distribution of ¯x?¯x-N(12, ?)
c. What is the probability that one randomly selected person learned to walk when the person was between 11.5 and 13.5 months old?
d. For the 13 people, find the probability that the average age that they learned to walk is between 11.5 and 13.5 months old?
e. Find the IQR for the average first time walking age for groups of 13 people.
Q1 = months
Q3 = months
IQR: months
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**b. Distribution of x̄:**
The distribution of the sample mean (x̄) is also normally distributed. The mean of the sample mean is the same as the population mean (μ = 12). The standard deviation of the sample mean (also called the standard error) is the population standard deviation (σ = 1.5) divided by the square root of the sample size (n = 13).
Standard Error = σ / √n = 1.5 / √13 ≈ 0.4160
So, x̄ ~ N(12, 0.4160)
**c. Probability for one person:**
We want to find P(11.5 < x < 13.5). First, convert these values to z-scores:
z₁ = (11.5 - 12) / 1.5 ≈ -0.3333
z₂ = (13.5 - 12) / 1.5 ≈ 1.0000
Now, look up these z-scores in a standard normal distribution table (or use a calculator or software) to find the corresponding probabilities:
P(z < -0.3333) ≈ 0.3694
P(z < 1.0000) ≈ 0.8413
P(11.5 < x < 13.5) = P(z < 1) - P(z < -0.3333) = 0.8413 - 0.3694 ≈ 0.4719
**d. Probability for the average of 13 people:**
Now we want to find P(11.5 < x̄ < 13.5). We use the same z-score formula, but with the standard error of the mean:
z₁ = (11.5 - 12) / 0.4160 ≈ -1.2019
z₂ = (13.5 - 12) / 0.4160 ≈ 1.2019
P(z < -1.2019) ≈ 0.1146
P(z < 1.2019) ≈ 0.8854
P(11.5 < x̄ < 13.5) = P(z < 1.2019) - P(z < -1.2019) = 0.8854 - 0.1146 ≈ 0.7708
**e. IQR for the average:**
To find the IQR, we need to find Q1 and Q3 for the distribution of x̄. We use the z-scores that correspond to the 25th and 75th percentiles (z = -0.6745 and z = 0.6745 respectively).
Q1 = μ + z * Standard Error = 12 + (-0.6745) * 0.4160 ≈ 11.7194
Q3 = μ + z * Standard Error = 12 + (0.6745) * 0.4160 ≈ 12.2806
IQR = Q3 - Q1 = 12.2806 - 11.7194 ≈ 0.5612
**Answers:**
b. x̄ ~ N(12, 0.4160)
c. 0.4719
d. 0.7708
e. Q1 = 11.7194 months, Q3 = 12.2806 months, IQR = 0.5612 months
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