Question

the english statistician karl pearson (1857-1936) introduced a formula for the skewness of a distribution. ( p = \frac{3(\bar{x} - \text{median})}{s} ) most distributions have an index of skewness between -3 and 3. when ( p > 0 ) the data are skewed right. when ( p < 0 ) the data are skewed left. when ( p = 0 ) the data are symmetric. calculate the coefficient of skewness for each distribution. describe the shape of each. (a) the coefficient of skewness for ( \bar{x} = 17 ), ( s = 2.2 ), median = 18 is ( p = -1.36 ). (round to the nearest hundredth as needed.) describe the shape of the distribution. a. the data are skewed left. b. the data are skewed right. c. the data are symmetric. (b) the coefficient of skewness for ( \bar{x} = 32 ), ( s = 5.6 ), median = 31 is ( p = square ). (round to the nearest hundredth as needed.)

Explanation:

Step1: Identify the formula

The formula for skewness \( P \) is \( P=\frac{3(\bar{x}-\text{median})}{s} \), where \( \bar{x} = 32 \), median \( = 31 \), and \( s = 5.6 \).

Step2: Substitute the values

Substitute \( \bar{x}=32 \), median \( = 31 \), and \( s = 5.6 \) into the formula:
\[
P=\frac{3(32 - 31)}{5.6}
\]

Step3: Simplify the numerator

First, calculate the numerator: \( 3(32 - 31)=3\times1 = 3 \).

Step4: Divide to find \( P \)

Now divide the numerator by the denominator: \( P=\frac{3}{5.6}\approx0.54 \) (rounded to the nearest hundredth).

Answer:

\( 0.54 \)