Question

Find \(\hat{p}\) and \(\hat{q}\) for each. a. \(X_{1}=25, n_{1}=75, X_{2}=40, n_{2}=90\) b. \(X_{1}=9, n_{1}=15, X_{2}=7, n_{2}=20\) c. \(X_{1}=3, n_{1}=20, X_{2}=5, n_{2}=40\) d. \(X_{1}=21, n_{1}=50, X_{2}=32, n_{2}=50\) e. \(X_{1}=20, n_{1}=150, X_{2}=30, n_{2}=50\)

Short answer

a. \(\hat{p} = 0.333\), \(\hat{q} = 0.444\); b. \(\hat{p} = 0.6\), \(\hat{q} = 0.35\); c. \(\hat{p} = 0.15\), \(\hat{q} = 0.125\); d. \(\hat{p} = 0.42\), \(\hat{q} = 0.64\); e. \(\hat{p} = 0.133\), \(\hat{q} = 0.6\).

Step-by-step solution

Understanding the Problem

You need to find \( \hat{p} \) and \( \hat{q} \) for each set of data. \( \hat{p} \) is the proportion of successes in one sample, while \( \hat{q} \) is the proportion of successes in the second sample.

Calculating \( \hat{p} \) and \( \hat{q} \) for Part a

To find \( \hat{p} \) for the first group, use the formula \( \hat{p} = \frac{X_{1}}{n_{1}} \). So for part a, \( \hat{p} = \frac{25}{75} = 0.333 \). To find \( \hat{q} \), use \( \hat{q} = \frac{X_{2}}{n_{2}} \). Therefore, \( \hat{q} = \frac{40}{90} \approx 0.444 \).

Calculating \( \hat{p} \) and \( \hat{q} \) for Part b

For part b, \( \hat{p} = \frac{9}{15} = 0.6 \), and \( \hat{q} = \frac{7}{20} = 0.35 \).

Calculating \( \hat{p} \) and \( \hat{q} \) for Part c

In part c, \( \hat{p} = \frac{3}{20} = 0.15 \), and \( \hat{q} = \frac{5}{40} = 0.125 \).

Calculating \( \hat{p} \) and \( \hat{q} \) for Part d

For part d, \( \hat{p} = \frac{21}{50} = 0.42 \), and \( \hat{q} = \frac{32}{50} = 0.64 \).

Calculating \( \hat{p} \) and \( \hat{q} \) for Part e

In this final part, \( \hat{p} = \frac{20}{150} \approx 0.133 \), and \( \hat{q} = \frac{30}{50} = 0.6 \).

Key concepts

Probability

Probability is a fascinating concept that tells us how likely something is to happen. It's like a way to measure chance. For example, if you're flipping a coin, the probability of getting heads is 50%. In mathematical terms, probability is usually expressed in decimal form between 0 and 1.

Think of it like this: a probability of 0 means an event will never happen, while a probability of 1 means it will definitely happen. Most events, however, have probabilities in between. This concept is crucial in predicting outcomes in games, experiments, and everyday life decisions.

When calculating probabilities, you often count how many ways an event can occur versus how many total possible events there are. For instance:

• If you have a bag with 3 red balls and 7 blue balls, the probability of drawing a red ball is calculated by dividing the number of red balls by the total number of balls: \( \text{Probability} = \frac{3}{10} = 0.3 \).

This basic idea helps in understanding more complex statistical concepts.

Success Rate

Knowing how often something successfully occurs, or the success rate, is key in many situations, from sports achievements to business outcomes. Success rate tells us how many times an event of interest occurs in comparison to all possible attempts.

In statistics, calculating the success rate is similar to determining proportions, as seen in the original exercise. We can represent this as:

• \( \text{Success Rate} = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Attempts}} \)

Success rates are especially useful for gauging efficiency and performance:

• When doctors test a new medication, the success rate helps them understand how effective it is.
• In manufacturing, tracking the success rate of defect-free products can improve quality control.

Understanding success rate gives insight into what works well and where improvements are needed.

Statistics

Statistics is the science of collecting, analyzing, and interpreting data. It's used in many different fields to make informed decisions and predictions.

Statistics involve understanding how data is distributed, calculating averages, and much more. It provides tools to find patterns and make sense of complex data sets. In our exercise, calculating proportions like \( \hat{p} \) and \( \hat{q} \) falls under basic statistical techniques.

Here are some fundamental aspects of statistics:

• Descriptive Statistics: Summarizes data using measures like mean, median, and mode.
• Inferential Statistics: Draws conclusions from data, like predicting trends or testing hypotheses.
• Data Visualization: Utilizes graphs and charts to represent data, making it easier to understand.

With statistics, you can reveal hidden patterns and insights that are not immediately obvious. Understanding statistics equips you with the skills to analyze and interpret various data forms.

Elementary Statistics

Elementary statistics is the foundation of understanding larger statistical concepts. It focuses on basic arithmetic, data collection, and simple analysis. This includes calculating averages, understanding probabilities, and finding proportions, as we've done with \( \hat{p} \) and \( \hat{q} \).

Here’s a brief look into what elementary statistics covers:

• Mean: The average value, found by adding all numbers in a data set and dividing by the number of numbers.
• Median: The middle number in a ordered data set, showing where the center most value lies.
• Range: The difference between the highest and lowest values, indicating spread.

Understanding these concepts helps form a solid foundation for more advanced statistical analysis. It's like learning the ABC's of data. By mastering elementary statistics, you're well-equipped to delve deeper into more complex statistical challenges later on.