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

Define the Proportion 3

The sample proportion
03 is defined as the ratio of the number of successes \(X\) to the total number of trials \(n\). It is given by the formula \(\hat{p} = \frac{X}{n}\).

Compute
03 and
0f for a

For the first case where \(n=52\) and \(X=32\), substitute these values into the formula for \(\hat{p}\): \[\hat{p} = \frac{32}{52} = 0.615\]. Then, calculate \(\hat{q}\) using \(\hat{q} = 1 - \hat{p}\): \[\hat{q} = 1 - 0.615 = 0.385\].

Compute
03 and
0f for b

For \(n=80\) and \(X=66\), substitute into the formula for \(\hat{p}\): \[\hat{p} = \frac{66}{80} = 0.825\]. Then, \(\hat{q}\) is \(\hat{q} = 1 - \hat{p}\): \[\hat{q} = 1 - 0.825 = 0.175\].

Compute
03 and
0f for c

For \(n=36\) and \(X=12\), compute \(\hat{p}\) as: \[\hat{p} = \frac{12}{36} = 0.333\]. Then find \(\hat{q}\): \[\hat{q} = 1 - 0.333 = 0.667\].

Compute
03 and
0f for d

For \(n=42\) and \(X=7\), calculate \(\hat{p}\): \[\hat{p} = \frac{7}{42} = 0.167\]. Then, \(\hat{q}\): \[\hat{q} = 1 - 0.167 = 0.833\].

Compute
03 and
0f for e

For \(n=160\) and \(X=50\), determine \(\hat{p}\): \[\hat{p} = \frac{50}{160} = 0.3125\]. Then compute \(\hat{q}\): \[\hat{q} = 1 - 0.3125 = 0.6875\].

Key concepts

Proportion Calculation

In statistics, a proportion is a special type of ratio that compares a part to a whole. It is often expressed as a fraction or a percentage. This is frequently used in scenarios where you want to understand how a part contributes to the entire set. For instance, if you're measuring how many students in a class of 100 like math, and 75 of them do, the proportion would be 75/100.

When we calculate proportions, the key components are the number of successes (or what you're counting) and the total number of trials (or the whole group size). The formula \(ackslash{hat{p} = \frac{X}{n}}\) where \(X\) is the count of your interest (success) and \(n\) is the total size of your group (trials), allows us to easily find the relative size of the part compared to the whole.

Sample Proportions

A sample proportion, represented as \(\hat{p}\), is a way to estimate the proportion from a larger population based on a smaller, representative sample. Imagine you want to know how popular a new movie is, but you only ask 50 people. If 30 like it, \(\hat{p}\) would be \(\frac{30}{50} = 0.6\) or 60%. This sample proportion helps in predicting the likely proportion for the entire population.

Sample proportions are central to inferential statistics because they allow us to make guesses about the larger population without needing information from everyone in it. This is crucial because it saves time and resources, while still providing a data-supported estimate. When you calculate a sample proportion, also remember \(\hat{q} = 1 - \hat{p}\) for the complement, which can be useful in understanding the rest of the dataset.

Statistical Notation

Statistical notation plays an essential role in the communication of quantitative data analyses. Notations like \(\hat{p}\) and \(n\) help keep our work precise and easy to follow. In context, \(\hat{p}\) denotes the sample proportion of successes, while \(n\) is the total number of observations or trials. Another important symbol is \(X\), which is the number of successes in your sample.

Using standardized notation is crucial because it makes mathematical expressions universally recognizable and readable. It avoids confusion when sharing findings with others, allowing for a seamless exchange of information in the statistical community. When diving into more complex topics, solid understanding of these notations will build your confidence and accuracy in problem-solving.

Successes and Trials

Successes and trials are terms used in binomial experiments or processes where the outcome is either a success or a failure. Here, a 'success' doesn't mean a positive outcome but the event or result we're interested in counting. The number of successes \(X\) and the number of trials \(n\) are the primary components to determine proportions.

In situations where you're calculating proportions, understanding what constitutes a success is crucial. If you surveyed 80 people, and 55 prefer chocolate ice cream (X), then a success would be each person saying they like chocolate, making \(X=55\). When these counts are plugged into the sample proportion formula, they allow us to derive insights into behavior or tendencies from specific populations. Understanding successes and trials will also aid in grasping advanced probability concepts later.