The lifespan of male mossquitoes are normally distributed with a mean of 6 days and a standard deviation of 7.5 hours. Calculate the upper and lower quartiles of the lifespan of these mosquitoes

To calculate the upper and lower quartiles of a normally distributed variable, we can use the **mean** and **standard deviation** of the distribution along with the properties of the normal distribution.

### Given:
- Mean \(\mu = 6 \, \text{days} = 144 \, \text{hours}\) (since 1 day = 24 hours)
- Standard deviation \(\sigma = 7.5 \, \text{hours}\)

The lower quartile (Q1) and upper quartile (Q3) correspond to the 25th and 75th percentiles of the distribution, respectively.

Using the standard normal distribution table:
- The z-score for the lower quartile (Q1) at the 25th percentile is approximately \(-0.674\)
- The z-score for the upper quartile (Q3) at the 75th percentile is approximately \(+0.674\)

### Step 1: Calculate the lower quartile (Q1)
The formula to convert a z-score to a value \(X\) in the original distribution is:

\[
X = \mu + (z \times \sigma)
\]

For Q1:
\[
Q1 = 144 + (-0.674 \times 7.5) = 144 - 5.055 = 138.945 \, \text{hours}
\]
So, \(Q1 \approx 138.95\) hours, which is approximately 5.79 days.

### Step 2: Calculate the upper quartile (Q3)
For Q3:
\[
Q3 = 144 + (0.674 \times 7.5) = 144 + 5.055 = 149.055 \, \text{hours}
\]
So, \(Q3 \approx 149.06\) hours, which is approximately 6.21 days.

### Summary:
- Lower quartile (Q1): 138.95 hours or 5.79 days
- Upper quartile (Q3): 149.06 hours or 6.21 days

Would you like a more detailed explanation or have any questions?

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**Related Questions:**
1. How would you calculate the median lifespan of these mosquitoes?
2. What is the probability that a mosquito lives less than 5 days?
3. How would the quartiles change if the standard deviation increased?
4. What is the lifespan range for the middle 50% of mosquitoes?
5. How can you interpret z-scores in real-world terms for this problem?

**Tip:** Remember that in a normal distribution, about 68% of the values fall within one standard deviation from the mean.

The lifespan of male mosquitoes is normally distributed with a mean of 6 days and a standard deviation of 7.5 hours. Calculate the upper and lower quartiles of the lifespan of these mosquitoes.

Discover how to calculate the upper and lower quartiles of a normally distributed lifespan, with a mean of 6 days and a standard deviation of 7.5 hours. Learn how to apply the normal distribution, z-scores, and percentiles for accurate statistical analysis. Suitable for students in Grades 10-12.

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