A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent...

1. Translate the problem requirements: We have a symmetric distribution around mean \(\mathrm{m}\), with 68% of data within one standard deviation (between \(\mathrm{m-d}\) and \(\mathrm{m+d}\)). We need to find what percent lies below the upper boundary \(\mathrm{m+d}\).
2. Apply symmetry properties: Use the fact that symmetric distributions split evenly around the mean to determine how the 68% is distributed on each side.
3. Account for the remaining distribution: Determine where the remaining 32% of data lies (outside the one standard deviation range) and how symmetry affects its placement.
4. Calculate cumulative percentage: Combine the percentage below the mean with the percentage between the mean and \(\mathrm{m+d}\) to find the total percentage below \(\mathrm{m+d}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in plain English. We're told about a population with some characteristic that follows a symmetric distribution. Think of this like a bell-shaped curve where the middle point is the mean \(\mathrm{m}\), and the curve looks exactly the same on both sides of this middle point.

We know that 68% of all people fall within one standard deviation \(\mathrm{d}\) of the mean. This means 68% of people have values between \((\mathrm{m} - \mathrm{d})\) and \((\mathrm{m} + \mathrm{d})\). Picture this as the "middle chunk" of our bell curve.

Our question asks: what percent of people have values less than \((\mathrm{m} + \mathrm{d})\)? In other words, if we draw a vertical line at the point \((\mathrm{m} + \mathrm{d})\), what percentage of the total population falls to the left of this line?

Process Skill: TRANSLATE - Converting the statistical language into a clear geometric picture

2. Apply symmetry properties

Now let's use the power of symmetry to break this down. Since our distribution is perfectly symmetric around the mean \(\mathrm{m}\), exactly half of all people fall below the mean, and exactly half fall above the mean. So 50% of people have values less than \(\mathrm{m}\).

Now, that middle chunk containing 68% of people is split equally on both sides of the mean because of symmetry. This means:

• 34% of people fall between \(\mathrm{m}\) and \((\mathrm{m} + \mathrm{d})\) [the right half of our middle chunk]
• 34% of people fall between \((\mathrm{m} - \mathrm{d})\) and \(\mathrm{m}\) [the left half of our middle chunk]

Think of it like cutting a symmetric sandwich in half - each side gets exactly the same amount.

Process Skill: INFER - Using symmetry to determine how the 68% splits around the mean

3. Account for the remaining distribution

We've accounted for 68% of people, but what about the remaining 32%? These are the people who fall outside our middle chunk - they're either much lower or much higher than the mean.

Again, symmetry helps us here. The remaining 32% is split equally:

• 16% of people have values less than \((\mathrm{m} - \mathrm{d})\) [far left tail]
• 16% of people have values greater than \((\mathrm{m} + \mathrm{d})\) [far right tail]

So our complete picture shows:

• 16% below \((\mathrm{m} - \mathrm{d})\)
• 34% between \((\mathrm{m} - \mathrm{d})\) and \(\mathrm{m}\)
• 34% between \(\mathrm{m}\) and \((\mathrm{m} + \mathrm{d})\)
• 16% above \((\mathrm{m} + \mathrm{d})\)

Notice how the left side (16% + 34% = 50%) perfectly mirrors the right side (34% + 16% = 50%).

4. Calculate cumulative percentage

Now we can answer our question: what percent of people have values less than \((\mathrm{m} + \mathrm{d})\)?

We need everyone to the left of the line at \((\mathrm{m} + \mathrm{d})\). This includes:

• The 16% who are below \((\mathrm{m} - \mathrm{d})\)
• The 34% who are between \((\mathrm{m} - \mathrm{d})\) and \(\mathrm{m}\)
• The 34% who are between \(\mathrm{m}\) and \((\mathrm{m} + \mathrm{d})\)

Adding these up: 16% + 34% + 34% = 84%

Technical notation summary: \(\mathrm{P(X < m + d) = 0.84 = 84\%}\)

Final Answer

Therefore, 84% of the distribution is less than \(\mathrm{m + d}\).

Looking at our answer choices, this corresponds to choice (D) 84%.

We can verify this makes sense: since \((\mathrm{m} + \mathrm{d})\) is one standard deviation above the mean, most people (but not all) should fall below this point, so 84% seems reasonable - it's much more than half (50%) but not nearly everyone (which would be close to 100%).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "within one standard deviation" as "exactly one standard deviation"

Students often confuse the phrase "within one standard deviation \(\mathrm{d}\) of the mean" to mean only the values exactly at \((\mathrm{m-d})\) and \((\mathrm{m+d})\), rather than understanding it means the entire range between \((\mathrm{m-d})\) and \((\mathrm{m+d})\). This leads them to incorrectly allocate the 68% and affects their entire solution approach.

2. Forgetting to account for the symmetry property

Even when students understand that the distribution is symmetric, they may fail to apply this crucial property when breaking down the 68%. They might not realize that symmetry means the 68% must be split equally (34% and 34%) on both sides of the mean, leading to incorrect percentage allocations.

3. Misunderstanding what "less than \(\mathrm{m + d}\)" means

Students may interpret "less than \(\mathrm{m + d}\)" as only including values between \(\mathrm{m}\) and \((\mathrm{m + d})\), forgetting that "less than" includes everything to the left of \((\mathrm{m + d})\) on the distribution. This fundamental misinterpretation leads to only considering part of the required area under the curve.

Errors while executing the approach

1. Incorrectly splitting the remaining 32% of the distribution

After identifying that 68% falls within one standard deviation, students may forget that the remaining 32% also splits symmetrically. They might assign the full 32% to one tail or split it unequally, leading to wrong calculations when determining cumulative percentages.

2. Arithmetic errors when adding percentages

When calculating the final answer, students may make simple addition errors. For example, they might calculate 16% + 34% + 34% incorrectly, or they may accidentally add wrong combinations of percentages (like adding the 16% from the right tail instead of using the correct segments).

Errors while selecting the answer

1. Choosing the percentage for the wrong region

Students may correctly calculate all the individual percentages but then select an answer that represents a different region. For instance, they might choose 16% (which represents the tail regions) or 32% (which represents both tails combined) instead of the correct cumulative percentage of 84%.