Question

The heights of a certain type of tree are approximately normally distributed with a mean height (μ) = 5ft and a standard deviation (σ) = 0.4ft. Which statement must be true?

1) The majority of trees will have a height less than 5ft.
2) The majority of trees will have a height greater than 5ft.
3) The majority of trees will have a height between 4.6ft and 5.4ft.
4) The majority of trees will have a height between 4.8ft and 5.2ft.

Answer 1

The statement that must be true is that the majority of trees will have a height between 4.6ft and 5.4ft.

Step-by-step explanation:

Using the information given, we have a normally distributed set of heights with a mean (μ) of 5ft and a standard deviation (σ) of 0.4ft. To determine which statement must be true, we can analyze the majority of trees' heights based on their probabilities.

1) The probability that a tree has a height less than 5ft is 0.5 since the mean is the same as the median in a normal distribution. Therefore, this statement is not necessarily true.

2) The probability that a tree has a height greater than 5ft is also 0.5. So, this statement is also not necessarily true.

3) To find the probability that a tree has a height between 4.6ft and 5.4ft, we can calculate the z-scores for these heights. Using the formula z = (x - μ) / σ, the z-score for 4.6ft is (4.6 - 5) / 0.4 = -1, and the z-score for 5.4ft is (5.4 - 5) / 0.4 = 1. Therefore, the probability is the area between these z-scores, which is approximately 0.6827. So, this statement is true.

4) Similarly, to find the probability that a tree has a height between 4.8ft and 5.2ft, we calculate the z-scores. The z-score for 4.8ft is (4.8 - 5) / 0.4 = -0.5, and the z-score for 5.2ft is (5.2 - 5) / 0.4 = 0.5. The probability is the area between these z-scores, which is approximately 0.3821. So, this statement is not true.