Problem 13 - Prealgebra

Question

As teenagers leave a concert, every 10th person is surveyed. They are asked if they would buy a T-shirt. One hundred forty out of a total of 800 people surveyed said yes. Is this sampling method valid? If so, how many people would you expect to buy T-shirts at the next concert if 7000 attend? Explain your reasoning.

Step-by-Step Solution

Answer

Yes, the method is valid; expect 1225 people to buy T-shirts.

Step 1: Identify the Sampling Method

The sampling method described in the exercise is a systematic sampling where every 10th person leaving the concert is surveyed. This is a valid sampling method if the list of attendees is randomly ordered or unordered.

Step 2: Calculate the Proportion of People Who Said Yes

To find the proportion of surveyed people who would buy a T-shirt, divide the number of people who said yes by the total number surveyed. This gives \( \frac{140}{800} = 0.175 \) or 17.5%.

Step 3: Apply the Proportion to the New Total

To find the expected number of people who would buy a T-shirt at the next concert, multiply the proportion by the total number of attendees. This means \( 0.175 \times 7000 = 1225 \).

Step 4: Consider the Validity of the Sampling

Systematic sampling is generally valid for estimating proportions as long as the order of selection is not correlated with the response. If the order is completely random or does not affect the likelihood of purchasing a T-shirt, the estimate should be reliable.

Key Concepts

Proportion Calculation

To understand proportion calculation, imagine you're trying to determine how many people in a group share a particular characteristic or behavior. For instance, when determining how many concert attendees would buy a T-shirt, you first survey a smaller group. If out of 800 people surveyed, 140 said they would buy a T-shirt, the proportion is found by dividing the number of "yes" responses by the total surveyed. This gives you:

• \( \frac{140}{800} = 0.175 \)

This means 17.5% of the respondents are expected T-shirt buyers. This percentage helps in predicting behaviors across a larger audience when conditions are similar. By calculating this proportion, you can apply it to any future events with similar audiences to estimate sales or interest, keeping in mind the calculated percentage from your sample.

Validity of Sampling Methods

Sampling methods are crucial when drawing conclusions from specific groups. One commonly used method is systematic sampling, where, for instance, every 10th person is surveyed. This method can be valid if appropriately applied. The key to validity in systematic sampling lies in the order. The method is reliable if:

• The list from which you select is in random or unordered order.
• The pattern of selection doesn't coincide with patterns in the population.

If the sequence of attendees doesn't influence their likelihood of buying a T-shirt, then the systematic sampling can be considered a valid reflection of the population. In other words, if you're surveying every 10th person as they leave, it works well unless there's a reason to believe every 10th person has different characteristics compared to the rest.

Expected Value Calculation

Expected value calculation helps estimate future outcomes based on historical data. It estimates how many people will, for example, buy a T-shirt if you know the proportion that has already expressed interest.Here's how you calculate the expected number:

• First, you find the proportion of interested buyers from the initial survey, as calculated earlier.
• Then, multiply this proportion by the estimated total number of future attendees. For example, if you expect 7000 people at the next concert, you apply the earlier found proportion: \( 0.175 \times 7000 = 1225 \).

This calculation suggests that approximately 1,225 attendees would likely purchase a T-shirt. Using expected value calculation provides a useful forecast when planning inventory or assessing potential demand at similar future events. This method is beneficial when you have partial data and need to project insights about a larger group.