Problem 1 - Algebra 2 and Trigonometry

Question

A seed company tests \(2,000\) tomato seeds and obtains \(1,954\) plants. The seed company advertises that their seed is 97\(\%\) productive. a. Is the seed company's claim justified? b. Is the seed company's claim based on theoretical or empirical probability?

Step-by-Step Solution

Answer

a. Yes, the claim is justified. b. It is based on empirical probability.

Step 1: Understand the Problem

We need to verify the seed company's claim that their seeds have a 97% productivity rate by comparing it with the actual results from the test they conducted.

Step 2: Calculate the Empirical Probability

The empirical probability of a seed growing into a plant is calculated by dividing the number of successful outcomes (plants obtained) by the total number of trials (seeds tested). Therefore, \( P(\text{plant}) = \frac{1954}{2000} \).

Step 3: Perform the Calculation

Calculate \( \frac{1954}{2000} = 0.977 \) or 97.7%. This is the actual productivity rate observed in the test.

Step 4: Compare with the Claimed Probability

The company claims a productivity rate of 97%. The empirical probability we calculated is 97.7%, which is higher than the claimed rate.

Step 5: Determine the Type of Probability Used

The company's claim is based on empirical probability because it is drawn from the actual experiment conducted, rather than a theoretical prediction.

Key Concepts

Probability in experiments

When we conduct experiments, particularly in fields like agriculture or manufacturing, we want to observe how often certain events occur. In the case of the seed company, the experiment consisted of planting 2,000 tomato seeds and counting how many successfully developed into plants. Some common terms used in these experiments. - **Trials**: Each instance or experiment performed, like planting one seed. In this scenario, there were 2,000 trials. - **Successful outcome**: An occurrence of the desired event, here it is a seed growing into a plant. - **Empirical probability**: This is determined by the actual results obtained from the experiment. For the seed company, the empirical probability of a seed turning into a plant is found by dividing the number of successful trials (1,954 plants) by the total number of trials (2,000 seeds tested), giving us a probability of 97.7%. This value helps determine whether their advertised claim is accurate.

Theoretical vs empirical probability

Theoretical and empirical probabilities are two approaches used to estimate the likelihood of an event occurring. - **Theoretical Probability**: This is based on expected outcomes derived from equally likely events without experiments. For instance, the theoretical probability of a fair coin landing heads is 0.5 because we know a coin has two equal sides. - **Empirical Probability**: Sometimes called experimental probability, this requires actual experiments for calculation. It is the probability that comes from what actually happens during an experiment rather than what should happen in theory. In the seed company's case, the claim was based on empirical probability, meaning they conducted an experiment to observe the conversion of seeds into plants and then used the data to make their claim. The empirical probability of 97.7% indicates real-world outcomes, which were tested by planting thousands of seeds.

Productivity rate calculation

Calculating productivity rates involves determining the success proportion in a series of trials. It provides insights into efficiency or effectiveness. In the context of the seed company, here's how it's done: - **Determine the number of successful outcomes**: Count how many seeds turned into plants, here it was 1,954. - **Determine the total number of trials**: Count the total seeds tested, which in this case was 2,000. - **Compute the productivity rate**: Divide the number of successful outcomes by the total trials and convert it into a percentage. Given the data, the productivity rate is calculated as \( \text{Productivity Rate} = \left( \frac{1954}{2000} \right) \times 100 \% = 97.7\% \) This rate suggests that most seeds result in successful plant growth, often more than the company's advertised 97%, supporting their seed viability claim.