Answer :
To solve this problem, we're going to find out the percentage of students who have a height of 176 cm or less, given that the average height of students is 160 cm with a standard deviation of 8 cm.
Here's how you break it down:
1. Understand the Problem:
- The average height (mean) of the students is 160 cm.
- The standard deviation, which measures the dispersion of the height values, is 8 cm.
- We want to find out what percentage of students have a height of 176 cm or less.
2. Calculate the Z-score:
- The Z-score tells us how many standard deviations away a particular value (in this case, 176 cm) is from the mean.
- The formula to calculate the Z-score for a value is:
[tex]\[
Z = \frac{\text{{value}} - \text{{mean}}}{\text{{standard deviation}}}
\][/tex]
- For the height of 176 cm:
[tex]\[
Z = \frac{176 - 160}{8} = \frac{16}{8} = 2.0
\][/tex]
3. Find the Probability:
- With the Z-score calculated, we can now find the probability of a student having a height less than or equal to 176 cm.
- This is done using the cumulative distribution function (CDF) for a standard normal distribution. The CDF gives the probability that a standard normal random variable will be less than or equal to a given value.
- For a Z-score of 2.0, the CDF value (or the probability) is approximately 97.72%.
4. Interpret the Result:
- Therefore, approximately 97.72% of students have a height of 176 cm or less.
So, about 97.72% of students are shorter than or equal to 176 cm, given the average height and the standard deviation provided.
Here's how you break it down:
1. Understand the Problem:
- The average height (mean) of the students is 160 cm.
- The standard deviation, which measures the dispersion of the height values, is 8 cm.
- We want to find out what percentage of students have a height of 176 cm or less.
2. Calculate the Z-score:
- The Z-score tells us how many standard deviations away a particular value (in this case, 176 cm) is from the mean.
- The formula to calculate the Z-score for a value is:
[tex]\[
Z = \frac{\text{{value}} - \text{{mean}}}{\text{{standard deviation}}}
\][/tex]
- For the height of 176 cm:
[tex]\[
Z = \frac{176 - 160}{8} = \frac{16}{8} = 2.0
\][/tex]
3. Find the Probability:
- With the Z-score calculated, we can now find the probability of a student having a height less than or equal to 176 cm.
- This is done using the cumulative distribution function (CDF) for a standard normal distribution. The CDF gives the probability that a standard normal random variable will be less than or equal to a given value.
- For a Z-score of 2.0, the CDF value (or the probability) is approximately 97.72%.
4. Interpret the Result:
- Therefore, approximately 97.72% of students have a height of 176 cm or less.
So, about 97.72% of students are shorter than or equal to 176 cm, given the average height and the standard deviation provided.
