The weights of adult long-tongued fruit bats are known to be normally distributed with a mean of 20.22 grams and a standard deviation of 3.23 grams. What is the probability that a randomly selected bat will:
a) weigh at most 15 grams?
b) weigh less than 30 grams?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to those Z scores.
A. :D
To find the probability regarding the weights of adult long-tongued fruit bats, we can use the standard normal distribution table or z-table. However, since we are given the mean and standard deviation of the weight distribution, we can transform these values to the standard normal distribution using z-scores and then find the probabilities.
A z-score represents the number of standard deviations an individual value is from the mean of a distribution. The formula to calculate the z-score is:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value of interest
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Now let's calculate the probabilities for each scenario:
a) Weigh at most 15 grams:
To find the probability that a random bat weighs at most 15 grams, we need to find the area under the normal distribution curve to the left of 15 grams.
First, calculate the z-score:
z = (15 - 20.22) / 3.23
z ≈ -1.61
Using the z-table, we can find the probability associated with the z-score of -1.61, which is approximately 0.053 (or 5.3%). Therefore, the probability that a randomly selected bat weighs at most 15 grams is 0.053.
b) Weigh less than 30 grams:
To find the probability that a random bat weighs less than 30 grams, we need to find the area under the normal distribution curve to the left of 30 grams.
Calculate the z-score:
z = (30 - 20.22) / 3.23
z ≈ 3.02
Using the z-table again, the probability associated with a z-score of 3.02 is approximately 0.998 (or 99.8%). Thus, the probability that a randomly selected bat weighs less than 30 grams is 0.998 (or 99.8%).
By utilizing the z-scores and the corresponding probabilities from the z-table, we can find the probabilities associated with specific weight ranges for the long-tongued fruit bats.