The heights of 10000 individuals are approximately normally distributed with a mean of 70 inches and a standard deviation of 3 inches. Find the probability that a person picked at random from this group will be between 65 and 74 inches tall.
To find the probability that a person picked at random from this group will be between 65 and 74 inches tall, we can use the z-score formula.
The z-score formula calculates the number of standard deviations a given value is from the mean. It is calculated using the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value we want to find the probability for
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
In this case, we want to find the probability of a person being between 65 and 74 inches tall. We can convert these values to z-scores and then use a z-table or a calculator to find the probabilities associated with those z-scores.
First, let's calculate the z-score for the lower bound (65 inches):
z_lower = (65 - 70) / 3 = -1.67
Next, let's calculate the z-score for the upper bound (74 inches):
z_upper = (74 - 70) / 3 = 1.33
Once we have the z-scores, we can find the probabilities associated with those z-scores using a z-table or a calculator.
Using a z-table, we can find the area (probability) between these two z-scores. Subtracting the area to the left of the lower bound from the area to the left of the upper bound gives us the probability of the person's height being between 65 and 74 inches.
Alternatively, if you have access to a statistics calculator or software, you can simply input the z-scores and find the probability directly.
Hope that helps! Let me know if you have any further questions.