Final answer:
The z-score for a value X=40, given a mean (μ) of 52 and a standard deviation (σ) of 15, is calculated to be -0.80. This z-score indicates that X is 0.80 standard deviations below the mean.
Step-by-step explanation:
To find the z-score for a given value, we use the formula:
z = (X - μ) / σ
Where X is the value in question, μ is the mean, and σ is the standard deviation. Given that μ=52, σ=15, and X=40, the z-score calculation would look like this:
z = (40 - 52) / 15
z = -12 / 15
z = -0.80
Thus, the z-score for X=40 is -0.80. This tells us that 40 is 0.80 standard deviations below the mean.
In the context of the normal distribution, a z-score of -0.80 suggests that the value X is within the first standard deviation to the left of the mean μ. This corresponds closely with the empirical rule, which states that about 68 percent of the x values lie within one standard deviation of the mean.