The average middle-distance runner at a local high school runs the mile in 4.5 minutes, with astandard deviation of 0.3 minute. What is the probability that a runner will run the mile in less than 4 minutes?
Possible Answers:
A)7%B)3%C)5%D)4%
I know it's C, but I don't know how to get it.
Can someone please show me a step by step solution, please. I am completely lost.
4 min is .5 min below the mean ... 1.667 standard deviations
-1.667 is called the z-score
a z-score table will show the fraction of the population at or below this point
the table actually shows about 4.8%
Best normal distribution applet on line:
http://davidmlane.com/normal.html
enter:
mean : 4.5
SD : 0.5
click on below, enter 4 to get .047799..
or 4.8% as Scott stated.
To calculate the probability that a runner will run the mile in less than 4 minutes, you can use the concept of Z-scores and the standard normal distribution.
Step 1: Determine the Z-score
The Z-score measures the number of standard deviations that a value is from the mean. In this case, we want to calculate the Z-score for the runner's time of 4 minutes using the formula:
Z = (X - μ) / σ
Where:
X = 4 minutes (the value we are interested in)
μ = mean (average time) = 4.5 minutes
σ = standard deviation = 0.3 minute
Plugging the values into the formula gives:
Z = (4 - 4.5) / 0.3 = -1.67
Step 2: Determine the probability using the Z-score
Once we have the Z-score, we can use a Z-table or a calculator to find the corresponding probability. In this case, we want to find the probability of a Z-score less than -1.67, which corresponds to the area under the standard normal distribution curve to the left of -1.67.
Using a Z-table or a calculator, we find that the probability for a Z-score of -1.67 is approximately 0.0475 or 4.75%.
Therefore, the probability that a runner will run the mile in less than 4 minutes is approximately 4.75%.
Since none of the given answer choices match exactly, the closest option is C) 5%.
To find the probability that a runner will run the mile in less than 4 minutes, we can use the concept of z-scores and the standard normal distribution.
Step 1: Convert the given values to z-scores.
In this case, we have the average time as 4.5 minutes with a standard deviation of 0.3 minutes. To find the z-score for a time of 4 minutes, we can use the formula:
z = (X - μ) / σ
where X is the given value, μ is the mean, and σ is the standard deviation.
z = (4 - 4.5) / 0.3
= -0.5 / 0.3
= -1.67
Step 2: Look up the z-score in the standard normal distribution table.
The z-score of -1.67 corresponds to a probability of 0.0475 (approximately) in the standard normal distribution table.
Step 3: Determine the probability of the runner running the mile in less than 4 minutes.
Since the question asks for the probability of running the mile in less than 4 minutes, we need to find the area under the curve to the left of the z-score. This is equivalent to the probability of the runner being faster than 4 minutes.
P(Z < -1.67) = 0.0475
However, we want the probability that the runner will be faster, so we subtract this probability from 1.
P(X < 4) = 1 - 0.0475
= 0.9525
Step 4: Convert the probability to a percentage.
To convert the probability to a percentage, we multiply by 100.
P(X < 4) = 0.9525 * 100
= 95.25%
Therefore, the probability that a runner will run the mile in less than 4 minutes is approximately 95.25%.
Since none of the provided answer choices match, none of them are correct.