Question

Suppose that the random selection process of numbers between 1 and 100 was conducted for a group of 100 values and for a group of 1,000 values. After the selection process is completed, the range of each group is determined. What would most likely be closest to the difference between each group's range of values? (A) 0 (B) 20 (C) 50 (D) 100

Short answer

The difference in range between the two groups is most likely 0.

Step-by-step solution

Understanding the Range

The range of a set of numbers is the difference between the largest and smallest numbers in the set. For numbers between 1 and 100, the maximum possible range is 100 - 1 = 99.

Calculating the Expected Range for 100 Values

When randomly selecting 100 values from 1 to 100, it's likely that both the smallest number (close to 1) and the largest number (close to 100) will be included in the set, giving a range close to 99.

Calculating the Expected Range for 1,000 Values

Similarly, when selecting 1,000 values from 1 to 100, the high number of selections will almost certainly cover nearly the entire range from 1 to 100, resulting also in a range close to 99.

Comparing the Ranges

Since both groups, one with 100 values and one with 1,000 values, are likely to span the full range from 1 to 100, their ranges will both be close to 99. Thus, the difference in ranges between the two groups is approximately 0.

Key concepts

Understanding Range Calculation

When it comes to determining how spread out values are within a set, the concept of range becomes invaluable. Range is calculated by subtracting the smallest number from the largest number within the set of numbers. This gives a clear indication of the span of the data points. For example, in a set of numbers between 1 and 100, the range would be calculated as:\[\text{Range} = \text{Largest Number} - \text{Smallest Number} = 100 - 1 = 99\]It's important to remember that the range doesn't provide information about how the numbers are distributed between the smallest and largest values. It's only about the extremes. Understanding this concept is especially handy in scenarios like the PSAT math problems, where knowing how to quickly determine the range can save you valuable time during the test.

Grasping Random Selection

Random selection refers to a process where each member of a set has an equal chance of being chosen. In the context of our number selection exercise, it implies that every number between 1 and 100 has an equal probability of being selected during each draw. This concept is crucial as randomness affects the likelihood of different outcomes occurring. For instance, with large samples, such as selecting 1,000 numbers out of this set, it’s inevitable due to randomness that nearly every number from 1 to 100 will appear, ensuring a broad coverage. This comprehensive coverage results in a range close to the maximum possible. Random selection ensures variability and gives a good idea of potential outcomes when sampling from larger groups. Understanding this principle aids students in predicting outcomes in statistical problems, enhancing their problem-solving skills.

Exploring Number Sets

A number set is essentially a collection of numbers that we analyze. The properties of these sets, such as their range, average (mean), and distribution, can tell us a lot about them. Number sets can be finite or infinite, uniform or varied in spacing as well. Let's take a look at the original exercise as an example. We had two number sets: one made up of 100 values and another of 1,000 values, all drawn randomly from the numbers between 1 and 100. The task was to determine the likely difference between their ranges. What was observed here was that irrespective of 100 or 1,000 selections, both sets likely covered numbers close to 1 and 100, thereby, both the sets are highly likely to achieve almost the full spectrum reaching close to the range of 99. Understanding number sets enables students to anticipate and interpret outcomes based on different scenarios. Being comfortable with number sets is a great advantage in solving math problems efficiently.