Question

for this application problem, assume that μ = x̄ and that σ = 5. compute these z - scores for these two classifications: option a: someone would be classified with \dwarfism\ if their height is less than 147cm. using your z - scores, determine the proportion of people in the full population (not the sample!) that would fit into the aforementioned classifications.

Explanation:

Step1: Recall the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean and $\sigma$ is the standard deviation. Here, assume $\mu = \bar{x}$ (sample mean) and $\sigma = 5$, and $x = 147$.

Step2: Calculate the z - score

Substitute the values into the formula: $z=\frac{147-\mu}{5}$. Since we don't know the value of $\mu$, if we assume for the sake of using the normal - distribution table after getting the z - score, and we calculate $z=\frac{147 - \mu}{5}$. Let's assume $\mu$ is the mid - point of the distribution (if we consider the histogram, we might estimate it, but without more information, we'll just use the formula). If we assume a normal distribution and use the standard normal table (z - table), once we have the z - score $z$, the proportion of the population that has a z - score less than the calculated $z$ is $P(ZLet's assume $\mu = 170$ (a rough estimate from the histogram's center). Then $z=\frac{147 - 170}{5}=\frac{- 23}{5}=-4.6$.

Step3: Find the proportion from the z - table

Looking up the z - score of $z=-4.6$ in the standard normal table, the proportion of values less than this z - score is approximately $0$. In a standard normal distribution table, the table usually goes from $z=- 3.49$ to $z = 3.49$, and for $z=-4.6$, the area to the left (the proportion of the population with height less than 147 cm) is extremely small and can be approximated as $0$.

Answer:

The proportion of people in the full population with height less than 147 cm is approximately $0$.