Assume that a school district has 10,000 6th graders. In this district, the average weight of a 6th grader is 80 pounds, with a standard deviation of 20 pounds. Suppose you draw a random sample of 50 students.
- What is the mean and standard deviation of the sampling distribution of the sample mean of samples of 50 students?
- What is the probability that the average weight of a sampled student will be less than 75 pounds?
- What is the probability that the average weight of a sampled student will be more than 75 pounds?
- What is the probability that the average weight of a sampled student will be between 70 and 90 pounds?
The Correct Answer and Explanation is :
Let’s break down each part of the problem:
1. Mean and Standard Deviation of the Sampling Distribution of the Sample Mean
The mean of the sampling distribution of the sample mean is equal to the population mean, and the standard deviation of the sampling distribution (also called the standard error) is given by the formula:
[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
]
Where:
- (\sigma = 20) (standard deviation of the population)
- (n = 50) (sample size)
Thus:
- Mean of the sampling distribution = 80 (same as the population mean)
- Standard deviation of the sampling distribution (standard error):
[
\text{Standard Error} = \frac{20}{\sqrt{50}} \approx 2.83
]
2. Probability that the Average Weight of a Sampled Student is Less than 75 Pounds
We need to find the probability that the sample mean is less than 75. We will first standardize this using the Z-score formula:
[
Z = \frac{X – \mu}{\text{Standard Error}}
]
Where:
- (X = 75) (the value we are interested in)
- (\mu = 80) (mean of the sampling distribution)
- (\text{Standard Error} = 2.83)
[
Z = \frac{75 – 80}{2.83} \approx -1.77
]
Now, using the Z-table, the probability corresponding to (Z = -1.77) is approximately 0.0384. So, the probability that the average weight of a sampled student is less than 75 pounds is about 0.0384 or 3.84%.
3. Probability that the Average Weight of a Sampled Student is More Than 75 Pounds
The probability of the average weight being more than 75 pounds is the complement of the probability of it being less than 75 pounds. Therefore:
[
P(X > 75) = 1 – P(X < 75) = 1 – 0.0384 = 0.9616
]
So, the probability that the average weight of a sampled student is more than 75 pounds is about 0.9616 or 96.16%.
4. Probability that the Average Weight of a Sampleed Student is Between 70 and 90 Pounds
For this, we will calculate the Z-scores for both 70 and 90, and then find the area between those two Z-scores.
First, calculate the Z-scores:
For (X = 70):
[
Z = \frac{70 – 80}{2.83} \approx -3.54
]
For (X = 90):
[
Z = \frac{90 – 80}{2.83} \approx 3.54
]
Using the Z-table:
- The probability for (Z = -3.54) is approximately 0.0002.
- The probability for (Z = 3.54) is approximately 0.9998.
The probability that the average weight is between 70 and 90 pounds is:
[
P(70 < X < 90) = P(Z = 3.54) – P(Z = -3.54) = 0.9998 – 0.0002 = 0.9996
]
Thus, the probability is about 0.9996 or 99.96%.
Summary of Results:
- Mean of the sampling distribution: 80
- Standard deviation of the sampling distribution: 2.83
- Probability that the average weight is less than 75 pounds: 3.84%
- Probability that the average weight is more than 75 pounds: 96.16%
- Probability that the average weight is between 70 and 90 pounds: 99.96%
Now, I’ll generate an image to visualize the probabilities based on the Z-scores for better understanding.
The graph above visualizes the normal distribution of the sample mean weights for 6th graders. Here’s a breakdown of the shaded regions:
- Red region: Represents the probability that the sample mean weight is less than 75 pounds (about 3.84%).
- Green region: Represents the probability that the sample mean weight is greater than 75 pounds (about 96.16%).
- Yellow region: Represents the probability that the sample mean weight falls between 70 and 90 pounds (about 99.96%).
This visualization shows how the probabilities are distributed based on the sample mean and the Z-scores.
