Question

the english statistician karl pearson (1857-1936) introduc
$p = \frac{3(\bar{x} - \text{median})}{s}$
most distributions have an index of skewness between $-$
the data are skewed left. when $p = 0$ the data are symme
distribution. describe the shape of each.
(a) the coefficient of skewness for $\bar{x} = 18$, $s = 2.8$, media
(round to the nearest hundredth as needed.)
describe the shape of the distribution.
\bigcirc a. the data are skewed left.
\bigcirc b. the data are skewed right.
\bigcirc c. the data are symmetric.

Explanation:

Step1: Identify the missing median value (assuming a typo, let's assume median is 16 as a common case for such problems, or maybe the original had a typo. Wait, maybe the user missed the median value. Wait, looking at the problem, maybe the median is 16? Wait, no, let's check again. Wait, the problem says "median" but the text is cut. Wait, maybe it's a common problem where mean is 18, median is 16, s=2.8. Let's proceed with that assumption (since otherwise we can't solve). So mean $\bar{x}=18$, median =16, s=2.8.

Step2: Plug into the skewness formula $P = \frac{3(\bar{x} - \text{median})}{s}$

Substitute the values: $\bar{x}=18$, median=16, s=2.8. So $3(18 - 16) = 3(2) = 6$. Then divide by s: $\frac{6}{2.8} \approx 2.14$? Wait, no, that can't be. Wait, maybe median is 17? Wait, no, maybe the original problem has median=17? Wait, no, perhaps the user made a typo. Wait, alternatively, maybe the median is 18? No, then P would be 0. Wait, maybe the median is 17. Let's check again. Wait, the problem as given: "The coefficient of skewness for $\bar{x} = 18$, $s = 2.8$, media" – maybe "media" is "median" with a typo, and the median is 17? Wait, no, let's think again. Wait, maybe the median is 16. Let's proceed with median=16.

So Step1: Calculate $\bar{x} - \text{median} = 18 - 16 = 2$

Step2: Multiply by 3: $3 \times 2 = 6$

Step3: Divide by s: $\frac{6}{2.8} \approx 2.14$? No, that's too high. Wait, maybe the median is 17.5? No, this is confusing. Wait, maybe the original problem has median=17. Let's try median=17. Then $\bar{x} - \text{median} = 1$, 3*1=3, 3/2.8≈1.07. Still positive. Wait, but the options: if P is positive, skewed right (since mean > median, skewed right). Wait, maybe the median is 17. Let's assume median=17 (maybe the user missed it). So:

Step1: Compute $\bar{x} - \text{median}$

$\bar{x} = 18$, median =17 (assumed), so $18 - 17 = 1$

Step2: Multiply by 3

$3 \times 1 = 3$

Step3: Divide by s

$s = 2.8$, so $\frac{3}{2.8} \approx 1.07$ (positive, so skewed right)

Wait, but maybe the median is 16. Let's check: 18-16=2, 3*2=6, 6/2.8≈2.14 (still positive, skewed right). If median is 18, then P=0 (symmetric). But the options include B (skewed right) and A (left) and C (symmetric). So if P is positive, skewed right (since mean > median, tail on right). So regardless of the median (as long as mean > median), P is positive, so skewed right.

Wait, maybe the median is 16 (common problem: mean=18, median=16, s=2.8). Let's recalculate:

Step1: $\bar{x} - \text{median} = 18 - 16 = 2$

Step2: $3 \times 2 = 6$

Step3: $\frac{6}{2.8} \approx 2.14$ (positive, so skewed right)

So the shape is skewed right, so option B.

Answer:

B. The data are skewed right.