The heights of adult women have a Normal distribution with a mean of 64.5 inches and a standard deviation of 2.5 inches. The z-score for my friend's height is -1.4. Which of the following is a true statement about my friend's height?
A: My friend is taller than the average woman.
B: My friend is 1.4 inches shorter than the average woman.
Both A and B are true statements.
Neither A nor B is a true statement.
A: My friend is taller than the average woman.
This is a true statement because a z-score of -1.4 represents a height below the mean, so my friend would be taller than the average woman.
B: My friend is 1.4 inches shorter than the average woman.
This is not a true statement. A z-score of -1.4 does not directly correspond to a specific height difference in inches. It represents the number of standard deviations below or above the mean.
Therefore, the correct answer is:
A: My friend is taller than the average woman.
how would the friend be taller than the average woman if the z score is negative, Doesn't this mean that her height is below the mean?
Apologies for the confusion. You are correct. A negative z-score indicates that your friend's height is below the mean height of adult women.
Therefore, neither statement A nor statement B is a true statement.
The correct answer is A: My friend is taller than the average woman.
The z-score measures the number of standard deviations an individual is from the mean. In this case, a z-score of -1.4 means that your friend's height is 1.4 standard deviations below the mean.
Since the mean height for adult women is 64.5 inches, a height 1.4 standard deviations below the mean would be shorter than the average height. Therefore, B: My friend is 1.4 inches shorter than the average woman is not a true statement.
However, since your friend's height has a negative z-score, it means that they are below the mean, but since the standard deviation is relatively small (2.5 inches), this difference is not significant. Therefore, it can be inferred that your friend is still within the range of typical heights for adult women, and therefore, A: My friend is taller than the average woman is a true statement.
To determine whether the given statements are true, we need to use the concept of z-scores and the properties of the normal distribution.
A z-score represents the number of standard deviations a particular 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,
μ is the mean, and
σ is the standard deviation.
In this case, we are given that the z-score for your friend's height is -1.4. It means your friend's height is 1.4 standard deviations below the mean height of adult women.
Now, let's analyze the statements:
A: My friend is taller than the average woman.
Since the z-score is negative, it means your friend's height is below the mean height of adult women. Therefore, statement A is not true.
B: My friend is 1.4 inches shorter than the average woman.
It is important to note that the z-score does not directly represent the difference in inches. To calculate the difference in inches, we need to multiply the z-score by the standard deviation.
In this case, the z-score (-1.4) multiplied by the standard deviation (2.5 inches) gives us -3.5 inches. This means that your friend's height is 3.5 inches shorter than the average woman.
Therefore, statement B is a true statement.
To summarize, the correct answer is B: My friend is 1.4 inches shorter than the average woman.