In a recent survey, twenty families reported their incomes for 1995. Was the range of reported 1995 incomes for these...

Understanding the Question

We have 20 families who reported their 1995 incomes. We need to determine if the range of these incomes exceeds \(\$60,000\).

Let's clarify what this means:

• Range = Maximum income - Minimum income
• We need a definitive yes/no answer: Is the range > \(\$60,000\)?

Key insight: For sufficiency, we must be able to answer either YES (the range is definitely greater than \(\$60,000\)) or NO (the range is definitely not greater than \(\$60,000\)). If different scenarios lead to different answers, then we don't have sufficient information.

Our approach: We'll test different possible income distributions to see if they lead to different answers about whether the range exceeds \(\$60,000\).

Analyzing Statement 1

Statement 1 tells us: Thirteen of the reported incomes were between \(\$20,000\) and \(\$35,000\).

This means 13 families have incomes in the range [\(\$20,000\), \(\$35,000\)]. But what about the remaining 7 families? Their incomes could be anywhere.

Let's test different scenarios:

Scenario 1 - All 7 remaining incomes are below \(\$20,000\):

• Suppose the lowest of these 7 incomes is \(\$10,000\)
• The highest overall income would be \(\$35,000\) (from the given 13)
• Range = \(\$35,000 - \$10,000 = \$25,000\)
• Is \(\$25,000 > \$60,000\)? NO

Scenario 2 - All 7 remaining incomes are above \(\$35,000\):

• Suppose the highest of these 7 incomes is \(\$100,000\)
• The lowest overall income would be \(\$20,000\) (from the given 13)
• Range = \(\$100,000 - \$20,000 = \$80,000\)
• Is \(\$80,000 > \$60,000\)? YES

Since we can construct scenarios that give us different answers (YES and NO), Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: Seven of the reported incomes were between \(\$80,000\) and \(\$95,000\).

This means 7 families have incomes in the range [\(\$80,000\), \(\$95,000\)]. But what about the remaining 13 families?

Let's test scenarios:

Scenario 1 - All 13 remaining incomes are far below \(\$80,000\):

• Suppose the lowest of these 13 incomes is \(\$10,000\)
• The highest overall income would be \(\$95,000\) (from the given 7)
• Range = \(\$95,000 - \$10,000 = \$85,000\)
• Is \(\$85,000 > \$60,000\)? YES

Scenario 2 - All 13 remaining incomes are just below \(\$80,000\):

• Suppose all 13 incomes are exactly \(\$79,000\)
• The highest overall income would be \(\$95,000\)
• The lowest would be \(\$79,000\)
• Range = \(\$95,000 - \$79,000 = \$16,000\)
• Is \(\$16,000 > \$60,000\)? NO

Again, we can get both YES and NO answers, so Statement 2 alone is NOT sufficient.

This eliminates choice B.

Combining Statements

Now let's use both statements together.

From both statements combined:

• 13 families have incomes between \(\$20,000\) and \(\$35,000\)
• 7 families have incomes between \(\$80,000\) and \(\$95,000\)
• Total families accounted for: \(13 + 7 = 20\) ✓

Important realization: This accounts for ALL 20 families! There are no incomes outside these two ranges.

But even knowing exactly where all 20 incomes fall, can we determine if the range exceeds \(\$60,000\)?

Testing the extremes:

Maximum possible range:

• Minimum income: \(\$20,000\) (lowest possible in the lower group)
• Maximum income: \(\$95,000\) (highest possible in the upper group)
• Range = \(\$95,000 - \$20,000 = \$75,000\)
• Is \(\$75,000 > \$60,000\)? YES

Minimum possible range:

• Minimum income: \(\$35,000\) (if all lower group incomes are at the top of their range)
• Maximum income: \(\$80,000\) (if all upper group incomes are at the bottom of their range)
• Range = \(\$80,000 - \$35,000 = \$45,000\)
• Is \(\$45,000 > \$60,000\)? NO

The range could be as large as \(\$75,000\) (YES) or as small as \$45,000\) (NO). Even with both statements together, we cannot definitively answer whether the range exceeds \(\$60,000\).

Therefore, the statements together are NOT sufficient.

This eliminates choice C.

The Answer: E

Even when we combine both statements and know exactly where all 20 incomes fall (in two specific, non-overlapping ranges), we still cannot determine whether the range exceeds \(\$60,000\). The range could be anywhere from \(\$45,000\) to \(\$75,000\), which includes values both above and below our \(\$60,000\) threshold.

Answer Choice E: "The statements together are not sufficient."