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?
Suppose that you select 2 cards without replacement from an ordinary deck of playing cards.
a) If the first card that you select is a heart, what is the probability that the second card that you select is
an heart?
b) If the first card that you select is an ace, what is the probability that the second card that you
select is a spade?
To find the probability in each scenario, we need to use the normal distribution and the z-score. The z-score measures the number of standard deviations a data point 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 we are interested in
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Now let's calculate the probability for each situation:
a) Weigh at most 15 grams:
To find the probability that a randomly selected bat will weigh at most 15 grams, we need to find the area under the normal distribution curve to the left of 15 grams.
First, we calculate the z-score:
z = (15 - 20.22) / 3.23
z = -1.61
Then, we look up the z-score in the standard normal distribution table or use a calculator to find the corresponding area. The calculated area represents the probability.
Using a standard normal distribution table, we find that the area to the left of z = -1.61 is approximately 0.0548. Hence, the probability that a randomly selected bat will weigh at most 15 grams is 0.0548 or 5.48%.
b) Weigh less than 30 grams:
To find the probability that a randomly selected bat will weigh less than 30 grams, we need to find the area under the normal distribution curve to the left of 30 grams.
First, we calculate the z-score:
z = (30 - 20.22) / 3.23
z = 3.01
Again, we look up the z-score in the standard normal distribution table or use a calculator to find the corresponding area, which represents the probability.
Using a standard normal distribution table, we find that the area to the left of z = 3.01 is approximately 0.9987. Hence, the probability that a randomly selected bat will weigh less than 30 grams is 0.9987 or 99.87%.
Therefore,
a) The probability that a randomly selected bat will weigh at most 15 grams is approximately 0.0548 or 5.48%.
b) The probability that a randomly selected bat will weigh less than 30 grams is approximately 0.9987 or 99.87%.