Question

a group of persons filled in a questionnaire their age and the number of states they visited. the results are recorded in the table below.
bivariate age and
number of states
visited

age\tstates visited
19\t5
20\t22
19\t17
36\t5
27\t7
52\t37
47\t14
53\t32
37\t22

-0.6
1.3
0.6

Explanation:

Step1: Identify Variables

Let \( x \) be Age (independent variable) and \( y \) be States Visited (dependent variable). The data points are: \((19,5)\), \((20,22)\), \((19,17)\), \((36,5)\), \((27,7)\), \((52,37)\), \((47,14)\), \((53,32)\), \((37,22)\).

Step2: Calculate Means

Mean of \( x \) (\(\bar{x}\)):
\(\bar{x} = \frac{19 + 20 + 19 + 36 + 27 + 52 + 47 + 53 + 37}{9}\)
\(= \frac{310}{9} \approx 34.44\)

Mean of \( y \) (\(\bar{y}\)):
\(\bar{y} = \frac{5 + 22 + 17 + 5 + 7 + 37 + 14 + 32 + 22}{9}\)
\(= \frac{161}{9} \approx 17.89\)

Step3: Calculate Covariance and Variances

Covariance (\(Cov(x,y)\)):
\(Cov(x,y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n - 1}\)

Variances:
\(Var(x) = \frac{\sum (x_i - \bar{x})^2}{n - 1}\)
\(Var(y) = \frac{\sum (y_i - \bar{y})^2}{n - 1}\)

Compute \((x_i - \bar{x})(y_i - \bar{y})\) for each point:

• \((19 - 34.44)(5 - 17.89) \approx (-15.44)(-12.89) \approx 199.02\)
• \((20 - 34.44)(22 - 17.89) \approx (-14.44)(4.11) \approx -59.35\)
• \((19 - 34.44)(17 - 17.89) \approx (-15.44)(-0.89) \approx 13.74\)
• \((36 - 34.44)(5 - 17.89) \approx (1.56)(-12.89) \approx -20.11\)
• \((27 - 34.44)(7 - 17.89) \approx (-7.44)(-10.89) \approx 81.02\)
• \((52 - 34.44)(37 - 17.89) \approx (17.56)(19.11) \approx 335.57\)
• \((47 - 34.44)(14 - 17.89) \approx (12.56)(-3.89) \approx -48.86\)
• \((53 - 34.44)(32 - 17.89) \approx (18.56)(14.11) \approx 261.98\)
• \((37 - 34.44)(22 - 17.89) \approx (2.56)(4.11) \approx 10.52\)

Sum of these products: \(199.02 - 59.35 + 13.74 - 20.11 + 81.02 + 335.57 - 48.86 + 261.98 + 10.52 \approx 873.53\)

\(Cov(x,y) = \frac{873.53}{8} \approx 109.19\)

Compute \(\sum (x_i - \bar{x})^2\):

• \((19 - 34.44)^2 \approx 238.4\)
• \((20 - 34.44)^2 \approx 208.5\)
• \((19 - 34.44)^2 \approx 238.4\)
• \((36 - 34.44)^2 \approx 2.43\)
• \((27 - 34.44)^2 \approx 55.35\)
• \((52 - 34.44)^2 \approx 308.35\)
• \((47 - 34.44)^2 \approx 157.75\)
• \((53 - 34.44)^2 \approx 344.5\)
• \((37 - 34.44)^2 \approx 6.55\)

Sum: \(238.4 + 208.5 + 238.4 + 2.43 + 55.35 + 308.35 + 157.75 + 344.5 + 6.55 \approx 1560.23\)
\(Var(x) = \frac{1560.23}{8} \approx 195.03\)

Compute \(\sum (y_i - \bar{y})^2\):

• \((5 - 17.89)^2 \approx 166.15\)
• \((22 - 17.89)^2 \approx 16.89\)
• \((17 - 17.89)^2 \approx 0.79\)
• \((5 - 17.89)^2 \approx 166.15\)
• \((7 - 17.89)^2 \approx 118.59\)
• \((37 - 17.89)^2 \approx 365.19\)
• \((14 - 17.89)^2 \approx 15.13\)
• \((32 - 17.89)^2 \approx 198.09\)
• \((22 - 17.89)^2 \approx 16.89\)

Sum: \(166.15 + 16.89 + 0.79 + 166.15 + 118.59 + 365.19 + 15.13 + 198.09 + 16.89 \approx 1063.97\)
\(Var(y) = \frac{1063.97}{8} \approx 132.996\)

Step4: Calculate Correlation Coefficient

Correlation coefficient \(r = \frac{Cov(x,y)}{\sqrt{Var(x) \cdot Var(y)}}\)
\(r = \frac{109.19}{\sqrt{195.03 \cdot 132.996}} \approx \frac{109.19}{\sqrt{25940.5}} \approx \frac{109.19}{161.06} \approx 0.68\), closest to \(0.6\).

Answer:

0.6