Math Problem Statement
Solution
We have a normal distribution with:
- Mean () = 80.2
- Standard deviation () = 93.6
- Sample size () = 50
The problem requires us to find two probabilities:
-
Probability that a single randomly selected value is between 67 and 73.6:
-
Probability that the mean of a sample of size is between 67 and 73.6:
Step-by-Step Solution
1. Probability for a Single Value (Question 1)
For a single value drawn from a normal distribution with mean and standard deviation , we standardize using the Z-score formula:
For :
For :
Calculate these Z-scores and then find the probability by finding the area between these two Z-scores.
2. Probability for the Sample Mean (Question 2)
When considering the sample mean of size , the distribution of the sample mean has a mean and a standard deviation given by:
Calculate :
Now, we standardize for the sample mean:
For :
For :
After calculating these Z-scores, find by looking up the area between the two Z-scores.
Let me perform these calculations for precise results.The calculated probabilities are:
-
Probability that a single randomly selected value is between 67 and 73.6:
-
Probability that the mean of a sample of size is between 67 and 73.6:
Would you like further explanation on any part of this solution?
Here are some follow-up questions for practice:
- What is the probability that a single randomly selected value is less than 67?
- What is the probability that a single randomly selected value is greater than 73.6?
- How would the probabilities change if the sample size increased to 100?
- What would be the probability if the range changed to ?
- Can you explain why the sample mean probability differs from the single value probability?
Tip: When working with normal distributions, remember that the standard deviation of the sample mean decreases as the sample size increases, which results in a narrower distribution for the sample mean.
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Math Problem Analysis
Mathematical Concepts
Statistics
Probability
Normal Distribution
Sampling Distribution
Formulas
Z = (X - μ) / σ for individual values
σ_M = σ / √n for sample mean
P(a < X < b) for continuous probabilities
Theorems
Central Limit Theorem
Suitable Grade Level
College Level
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