SOLUTION: Assume that the Weight of the students in your sample enrolled at NSU have an approximately Normal distribution. Find the probability that the mean Weight of your sample selected

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Question 1166864: Assume that the Weight of the students in your sample enrolled at NSU have an
approximately Normal distribution. Find the probability that the mean Weight of your
sample selected from the university is (Quantitative variable)
A. Within 2 standard deviations from the mean.
B. More than 1 standard deviation from the mean.

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2030) (Show Source):
You can put this solution on YOUR website!
The probability that the mean weight of your sample falls within a certain number of standard deviations from the population mean can be determined using the **Empirical Rule** (or 68-95-99.7 Rule) for a Normal distribution.
The problem implies that the distribution of the **sample mean** ($\bar{X}$) is also Normal (due to the original distribution being Normal), and we are measuring distances in terms of the standard deviation of the sample mean (the **Standard Error**, $\sigma_{\bar{X}}$).
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## A. Within 2 Standard Deviations from the Mean
The probability that the sample mean is within **2 standard deviations** of the population mean ($\mu \pm 2\sigma_{\bar{X}}$) is approximately **95%**.
* $P(\mu - 2\sigma_{\bar{X}} < \bar{X} < \mu + 2\sigma_{\bar{X}}) \approx \mathbf{0.95}$ or $\mathbf{95\%}$
This value comes directly from the Empirical Rule, which states that approximately 95% of the data in a Normal distribution lies within two standard deviations of the mean.
[Image of a normal distribution curve with the area within 2 standard deviations shaded]

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## B. More than 1 Standard Deviation from the Mean
The probability that the sample mean is **more than 1 standard deviation** from the mean is the area in the two tails combined (i.e., outside the range $\mu \pm 1\sigma_{\bar{X}}$).
1. **Find the area within 1 standard deviation:**
According to the Empirical Rule, the probability that the sample mean is **within 1 standard deviation** of the population mean is approximately **68%**.
$$P(\mu - 1\sigma_{\bar{X}} < \bar{X} < \mu + 1\sigma_{\bar{X}}) \approx 0.68$$
2. **Calculate the area outside 1 standard deviation:**
Since the total area under the probability curve is $1$ (or $100\%$), the probability of the sample mean being *more than* $1$ standard deviation away is $1$ minus the probability of it being *within* $1$ standard deviation.
$$\text{Probability (More than 1 SD away)} = 1 - P(\text{Within 1 SD})$$
$$\text{Probability (More than 1 SD away)} \approx 1 - 0.68$$
$$\text{Probability (More than 1 SD away)} \approx \mathbf{0.32}$$ or $\mathbf{32\%}$
* $P(|\bar{X} - \mu| > 1\sigma_{\bar{X}}) \approx \mathbf{0.32}$ or $\mathbf{32\%}$

Answer by ikleyn(53107) (Show Source):
You can put this solution on YOUR website!
.
Assume that the Weight of the students in your sample enrolled at NSU have an
approximately Normal distribution.
Find the probability that the mean Weight of your sample selected from the university is (Quantitative variable)
A. Within 2 standard deviations from the mean.
B. More than 1 standard deviation from the mean.
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I think that both the answers to both questions in the post by @CPhill are incorrect.

They had be correct, if the question would ask about the weight of an individual random student.

But the questions are DIFFERENT: they ask about the mean weight of a sample.
So, the answers should be different.