College

The weights of students in a Statistics class vary according to a normal distribution with a mean of 140 pounds and a standard deviation of 16 pounds.

Approximately what percentage of the students weigh more than 172 pounds?

Answer :

To find the percentage of students who weigh more than 172 pounds, we need to analyze the weights using the properties of the normal distribution.

Here's how you can find the answer:

1. Identify the Given Values:
- Mean (average) weight of students = 140 pounds
- Standard deviation = 16 pounds
- Weight threshold of interest = 172 pounds

2. Calculate the Z-score:
- The Z-score helps us understand how far a specific value is from the mean, in terms of standard deviations.
- Formula for the Z-score:
[tex]\[
Z = \frac{{\text{{Weight Threshold}} - \text{{Mean}}}}{\text{{Standard Deviation}}}
\][/tex]
- Plug in the values:
[tex]\[
Z = \frac{{172 - 140}}{16} = \frac{32}{16} = 2.0
\][/tex]

3. Find the Probability:
- Using the Z-score, we can find out the probability of a student weighing more than 172 pounds.
- The standard normal distribution table (or a calculator) tells us the probability associated with a Z-score of 2.0.
- For [tex]\(Z = 2.0\)[/tex], the probability that a student weighs more than 172 pounds is approximately 0.02275 (or 2.275%).

4. Interpret the Result:
- This means that approximately 2.275% of the students weigh more than 172 pounds.

Therefore, the percentage of students who weigh more than 172 pounds is about 2.275%.

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