A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 40 seconds. The fitness association wants to recognize the fastest 10% of the boys with certificates of recognition. What time would the boys need to beat in order to earn a certificate of recognition from the fitness association?
Choices:
a. 515.8 seconds
b.384.2 seconds
c.398.8 seconds
d.501.2 seconds
I got d 501.2 seconds is that correct?
correct.
To find the time that the boys would need to beat in order to earn a certificate of recognition from the fitness association, we can use the concept of z-scores.
A z-score measures the number of standard deviations a particular data point is from the mean of a distribution. In this case, we want to find the time that corresponds to the top 10% fastest boys. To do this, we need to find the z-score that corresponds to the top 10% of the normal distribution.
Step 1: Find the z-score corresponding to the top 10%.
The top 10% corresponds to the area under the normal curve to the right of the z-score. We can use a standard normal distribution table or a calculator to find that the z-score is approximately 1.28.
Step 2: Use the z-score formula to find the time.
The z-score formula is z = (x - μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation. Rearranging the formula, we have x = (z * σ) + μ.
Plugging in the values, we have:
x = (1.28 * 40) + 450
x = 51.2 + 450
x = 501.2 seconds
Therefore, the boys would need to beat a time of 501.2 seconds in order to earn a certificate of recognition from the fitness association. So your answer of d) 501.2 seconds is correct.