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 460 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) 408.8
b) 511.2
c) 394.2
d) 525.8

I got b. Is that correct?

Remember faster means less seconds not more so that means the answer can not be b because it would be the slowest time.

To find the time that the boys need to beat in order to earn a certificate of recognition, we need to determine the cutoff point that represents the fastest 10% of the boys.

First, we need to find the z-score corresponding to the cutoff point in the standard normal distribution. The z-score represents the number of standard deviations away from the mean.

Since the boys need to beat the cutoff point, we are interested in the upper tail of the distribution. The area under the normal curve to the left of the cutoff point is 1 - 0.10 = 0.90.

Using a standard normal distribution table or a calculator, we find that the z-score corresponding to an area of 0.90 to the left is approximately 1.28.

Next, we can use the z-score formula to find the corresponding value in the original distribution:

z = (x - μ) / σ

Where:
z is the z-score,
x is the desired value,
μ is the mean, and
σ is the standard deviation.

Plugging in the values, we have:

1.28 = (x - 460) / 40

Solving for x, we get:

1.28 * 40 + 460 = x
51.2 + 460 = x
511.2 = x

Therefore, the boys would need to beat a time of 511.2 seconds in order to earn a certificate of recognition.

So, the correct answer is b) 511.2.

To find the time that the boys would need to beat in order to earn a certificate of recognition, we can use the concept of z-scores, which represent the number of standard deviations a data point is from the mean in a normal distribution.

First, we need to find the z-score corresponding to the top 10% of the distribution. Since the normal distribution is symmetric, we can find the z-score using the standard normal distribution table or calculator.

The top 10% of the distribution corresponds to a z-score of approximately 1.28. This means that the time needed to earn a certificate of recognition is 1.28 standard deviations above the mean.

Next, we calculate the time by adding the z-score (1.28) multiplied by the standard deviation (40 seconds) to the mean (460 seconds):

Time = 460 + (1.28 * 40)
Time = 460 + 51.2
Time ≈ 511.2

Therefore, the answer is b) 511.2 seconds. You got it correct!

Right!