Question

Find the proportions \(\hat{p}\) and \(\hat{q}\) for each. a. \(n=52, X=32\) b. \(n=80, X=66\) c. \(n=36, X=12\) d. \(n=42, X=7\) e. \(n=160, X=50\)

Short answer

a. \(\hat{p} = 0.615\), \(\hat{q} = 0.385\); b. \(\hat{p} = 0.825\), \(\hat{q} = 0.175\); c. \(\hat{p} = 0.333\), \(\hat{q} = 0.667\); d. \(\hat{p} = 0.167\), \(\hat{q} = 0.833\); e. \(\hat{p} = 0.3125\), \(\hat{q} = 0.6875\).

Step-by-step solution

Understand the Definitions

The sample proportion \( \hat{p} \) is calculated as \( \hat{p} = \frac{X}{n} \), where \( X \) is the number of successes, and \( n \) is the total number of trials or observations. The complement proportion \( \hat{q} \) is \( \hat{q} = 1 - \hat{p} \). We will use these formulas to find the proportions for each part.

Calculate Proportions for part (a)

For \( n = 52 \) and \( X = 32 \), the sample proportion is \( \hat{p} = \frac{32}{52} \approx 0.615 \). Therefore, \( \hat{q} = 1 - 0.615 = 0.385 \).

Calculate Proportions for part (b)

For \( n = 80 \) and \( X = 66 \), the sample proportion is \( \hat{p} = \frac{66}{80} = 0.825 \). Thus, \( \hat{q} = 1 - 0.825 = 0.175 \).

Calculate Proportions for part (c)

For \( n = 36 \) and \( X = 12 \), the sample proportion is \( \hat{p} = \frac{12}{36} = 0.333 \). Therefore, \( \hat{q} = 1 - 0.333 = 0.667 \).

Calculate Proportions for part (d)

For \( n = 42 \) and \( X = 7 \), the sample proportion is \( \hat{p} = \frac{7}{42} \approx 0.167 \). Thus, \( \hat{q} = 1 - 0.167 = 0.833 \).

Calculate Proportions for part (e)

For \( n = 160 \) and \( X = 50 \), the sample proportion is \( \hat{p} = \frac{50}{160} = 0.3125 \). Therefore, \( \hat{q} = 1 - 0.3125 = 0.6875 \).

Key concepts

Understanding Complement Proportion

In statistics, the complement proportion is a concept that helps us see what part of the whole isn't represented by our primary data. When we measure something, like the number of people who prefer chocolate over vanilla, our initial focus is on chocolate lovers. But what about the vanilla fans? That's where the complement proportion, denoted as \(\hat{q}\), comes into play.

The complement proportion is calculated using the formula \(\hat{q} = 1 - \hat{p}\), where \(\hat{p}\) is the sample proportion representing the part of the dataset we are initially interested in. Think of \(\hat{q}\) as providing balance to your data analysis.

For instance, if 60% of people prefer chocolate, it means 40% (\(\hat{q} = 1 - 0.6 = 0.4\)) prefer other flavors, showing the full picture of tastes in your group. Understanding both the proportion \(\hat{p}\) and its complement \(\hat{q}\) allows for a more complete analysis of data.

Proportion Calculation Basics

Calculating proportions is a fundamental skill in statistics that helps you summarize large amounts of data. It's about understanding parts of a whole and using that insight to draw meaningful conclusions.

To compute the sample proportion \(\hat{p}\), you use the formula \(\hat{p} = \frac{X}{n}\), where \(X\) is the count of a specific outcome, like the number of successful trials, and \(n\) is the total number of trials.

Consider this scenario: You conducted a survey with 100 participants to find out who likes ice cream, and 70 said yes. The sample proportion \(\hat{p}\) is \(\frac{70}{100} = 0.7\), which means 70% of your surveyed population enjoy ice cream. This calculation simplifies large datasets into digestible pieces of information.

Basic Statistics and Proportions

Basic statistics revolve around summarizing and interpreting data to make informed conclusions about larger populations, and proportions play a pivotal role in this.

Proportions help us understand the distribution of elements within a group. For example, knowing that 75% of surveyed individuals prefer tea over coffee gives us insights into popular preferences, which a business might use to stock inventory or plan marketing strategies.

When using sample proportions, it's crucial to remember that they are estimates of the actual proportion in the entire population. This means while they are useful, they always have a degree of uncertainty. This is why in statistics, sample proportions are often used along with confidence intervals to gauge their precision and reliability.

Understanding proportions in statistics helps build foundational knowledge essential for more complex analyses and critical decision-making based on data.