Final answer:
A z-score is calculated using the formula z = (x - μ) / σ and is used to find the probability of a data point by referencing the area under the normal distribution curve in a z-table. The area under the curve to the left of the z-score represents the probability that a value is less than or equal to the corresponding z-score. An individual's height can also be computed by rearranging the z-score formula if the z-score is known.
Step-by-step explanation:
To calculate a z-score, you would use the formula z = (x - μ) / σ, where x is the value in question, μ (mu) is the average height, and σ (sigma) is the standard deviation. The z-score indicates how many standard deviations an element is from the mean.
Once you have the z-score, you can find the probability associated with this score by consulting a z-table, which shows the area under the normal distribution curve corresponding to various z-score values. This area represents the probability that a randomly selected individual will have a height less than or equal to that corresponding to the z-score.
If a 15-to-18-year-old male from Chile in 2009-2010 has a z-score of z = 1.27, this tells us that the male's height is 1.27 standard deviations to the right of the mean. To find the actual height, the z-score is substituted back into the formula, which is rearranged to x = μ + zσ.
The statement X~ (100, 15) translates to "The random variable X is normally distributed with an average (mean) of 100 and a standard deviation of 15."
For the World Health Organization example given, the distribution of height for girls aged five is denoted as X ~ N(109, 4.5). The z-score for a height of 112 cm is calculated as z = (112 - 109) / 4.5. Similarly, the z-score for heights of 100 cm and 105 cm can also be calculated using the same formula. To find the height that corresponds to a z-score of 1.5, you would use x = 109 + (1.5 × 4.5).
Using the empirical rule (68-95-99.7 rule), for the distribution of girls' height, approximately 68 percent of the values would be expected to lie within one standard deviation (109 ± 4.5 cm). This gives a range of values from 104.5 cm to 113.5 cm.
For an NBA player whose height is 85 inches, if the z-score formula gives us z = 1.5424, the height 85 inches is 1.5424 standard deviations above the mean, which indicates the player is taller than average.
In the case of the 15-to 18-year-old male from Chile being 176 cm tall, you would calculate the z-score and then determine if this height is above (to the right) or below (to the left) the mean based on whether the z-score is positive or negative.