Question

as part of a statistics project, a teacher brings a bag of marbles containing 500 white marbles and 200 red marbles. she tells the students the bag contains 700 total marbles, and asks her students to determine how many red marbles are in the bag without counting them.

a student randomly draws 100 marbles from the bag. of the 100 marbles, 25 are red.

the data collection method can best be described as

• survey
• clinical study
• census
• controlled study

the target population consists of

• the 25 red marbles drawn by the student
• the 100 marbles drawn by the student
• the 200 red marbles in the bag
• the 700 marbles in the bag
• none of the above

the sample consists of

• the 100 marbles drawn by the student
• the 25 red marbles drawn by the student
• the 700 marbles in the bag
• the 200 red marbles in the bag
• none of the above

based on the sample, the student would estimate that marbles in the bag were red.

Explanation:

⚡ Using what you learned: Population vs. Sample · Sampling Methods

Step 1: Identify the data collection method

The student is gathering data by selecting a subset of the items in the bag to gather information about the whole group.

• Survey: In statistics, a survey is a method of gathering data from a sample of individuals (or objects) in a population to make inferences about the entire population. Drawing a sample of marbles to estimate the total composition is a physical representation of a sample survey.
• Clinical study / Controlled study: These involve applying a treatment to experimental units to observe responses, which is not happening here.
• Census: A census requires collecting data from every single member of the population (all 700 marbles), which was not done.

Therefore, the data collection method is best described as a Survey.

Step 2: Identify the target population

The target population is the entire group of individuals or objects that we want to draw conclusions about.

In this scenario, the students are trying to determine how many red marbles are in the entire bag. The entire bag contains 700 marbles.

Therefore, the target population consists of The 700 marbles in the bag.

Step 3: Identify the sample

The sample is the specific subset of the population from which data is actually collected.

The student randomly draws 100 marbles from the bag to analyze them.

Therefore, the sample consists of The 100 marbles drawn by the student.

Step 4: Calculate the estimated number of red marbles

To estimate the total number of red marbles in the bag, we use the proportion of red marbles found in the sample and apply it to the entire population:

1. Find the sample proportion of red marbles:

$$ \text{Sample Proportion} = \frac{25}{100} = 0.25 $$

1. Multiply this proportion by the total population size (\(700\)):

$$ \text{Estimated Red Marbles} = 0.25 \times 700 = 175 $$

Based on the sample, the student would estimate that 175 marbles in the bag were red.

Answer:

• The data collection method can best be described as: Survey
• The target population consists of: The 700 marbles in the bag
• The sample consists of: The 100 marbles drawn by the student
• Based on the sample, the student would estimate that: 175