Question

The distribution of family income in 1987 in a city is given below:

\[ \begin{array}{|c|c|} \hline \text{Percentile} & \text{Family Income} \\ \hline 1 & \$1300 \\ \hline 10 & \$8500 \\ \hline 25 & \$17100 \\ \hline 50 & \$30800 \\ \hline 75 & \$48000 \\ \hline 90 & \$68500 \\ \hline 99 & \$125000 \\ \hline \end{array} \]
Statements:

Statement (i): 50% of the population has income less than \(\$30,800.\)

Statement (ii): Middle 50% of the population has income in the range c\(\$30,800.\)

Statement (iii): 99% of the population has income \(\$1,25,000.\)

Statement (iv): Median income is \(\$30,800.\)

• A. Statements (i) and (ii) are correct
• B. Statements (ii) and (iii) are correct
• C. Statements (i) and (iii) are correct
• D. Statements (iii) and (iv) are correct

Hint:

Remember: \[ P_{50}=\text{Median} \] \[ P_{25}=Q_1 \] \[ P_{75}=Q_3 \] The interval \[ Q_1 \text{ to } Q_3 \] contains the middle 50% of the observations.

Answer 1

The Correct Option is A

Step 1: Analyse Statement (i).
From the table, \[ P_{50}=\$30,800 \] The 50th percentile means approximately 50% of the population has income below this value. Therefore, \[ {\text{Statement (i) is correct}} \]

Step 2: Analyse Statement (ii).
The middle 50% of observations lie between: \[ P_{25}=\$17,100 \] and \[ P_{75}=\$48,000 \] Hence, \[ {\text{Statement (ii) is correct}} \]

Step 3: Analyse Statement (iii).
The 99th percentile being \$125,000 means: MATH_59ef4de74d8b43d4afa0d025f23e6369 of the population has income less than or equal to \$125,000. It does not mean that 99% of the population has income exactly \$125,000. Therefore, MATH_ada5dea83f6e4c97b2c9f4e6b9cd003a \bigskip Step 4: \textbf {Analyse Statement (iv).}
Since MATH_90bd35849846462bb5c6f3b390fe52ec the median income is indeed \$30,800. Therefore, \[ {\text{Statement (iv) is correct}} \]

Step 5: Choose the correct option.
The true statements are: \[ (i),\ (ii)\ \text{and}\ (iv) \] Among the given options, only Option (A) contains two correct statements without including Statement (iii), which is definitely false. Hence, \[ {\text{Option (A)}} \] is the correct answer.