The heights of Japanese maple trees are normally distributed with a mean of 6 meters and a standard deviation of 0.2 meter. What percent of Japanese maple trees are between 5.5 meters and 6.7 meters tall?
http://davidmlane.com/hyperstat/z_table.html
To find the percentage of Japanese maple trees that are between 5.5 and 6.7 meters tall, we can use the cumulative distribution function (CDF) of the normal distribution.
The CDF gives the probability that a random variable, in this case the height of Japanese maple trees, is less than or equal to a given value. To find the percentage between two values, we can subtract the CDF of the lower value from the CDF of the higher value.
First, we need to find the z-scores for the heights of 5.5 meters and 6.7 meters. The z-score tells us how many standard deviations away from the mean a particular value is.
For 5.5 meters:
z-score = (x - mean) / standard deviation
= (5.5 - 6) / 0.2
= -2.5
For 6.7 meters:
z-score = (x - mean) / standard deviation
= (6.7 - 6) / 0.2
= 3.5
Now, we can use the z-scores to find the probabilities using a standard normal distribution table or a calculator. The probability between these two z-scores will give us the percentage of Japanese maple trees between 5.5 and 6.7 meters tall.
Let's find the probabilities:
P(z < -2.5) = 0.0062 (from standard normal distribution table or calculator)
P(z < 3.5) = 0.9998 (from standard normal distribution table or calculator)
To get the percentage between the two heights, we subtract the lower probability from the higher probability:
P(5.5 < x < 6.7) = P(z < 3.5) - P(z < -2.5)
= 0.9998 - 0.0062
= 0.9936
Therefore, approximately 99.36% of Japanese maple trees are between 5.5 and 6.7 meters tall.