Question

Find \(\hat{p}\) and \(\hat{q}\) for each. a. \(n=36, X=20\) b. \(n=50, X=35\) c. \(n=64, X=16\) d. \(n=200, X=175\) e. \(n=148, X=16\)

Short answer

a) \(\hat{p} = \frac{5}{9}\), \(\hat{q} = \frac{4}{9}\) b) \(\hat{p} = 0.7\), \(\hat{q} = 0.3\) c) \(\hat{p} = 0.25\), \(\hat{q} = 0.75\) d) \(\hat{p} = 0.875\), \(\hat{q} = 0.125\) e) \(\hat{p} \approx 0.108\), \(\hat{q} \approx 0.892\)

Step-by-step solution

Understanding the Parameters

Firstly, we need to understand the given parameters. Here, \(n\) represents the total number of trials, and \(X\) represents the number of successful trials in a binomial setting. The tasks ask for the estimation of the sample proportion of success, denoted as \(\hat{p}\), and the sample proportion of failure, denoted as \(\hat{q}\), where \(\hat{q} = 1 - \hat{p}\).

Calculating \(\hat{p}\) for Each Case

We calculate \(\hat{p}\) using the formula \(\hat{p} = \frac{X}{n}\). This gives us the observed proportion of successes in our trials.

Case A: Calculate \(\hat{p}\) and \(\hat{q}\)

For a), \(n = 36\), \(X = 20\). \[\hat{p} = \frac{20}{36} = \frac{5}{9}\]. Then \(\hat{q} = 1 - \hat{p} = 1 - \frac{5}{9} = \frac{4}{9}\).

Case B: Calculate \(\hat{p}\) and \(\hat{q}\)

For b), \(n = 50\), \(X = 35\). \[\hat{p} = \frac{35}{50} = 0.7\]. Then \(\hat{q} = 1 - \hat{p} = 1 - 0.7 = 0.3\).

Case C: Calculate \(\hat{p}\) and \(\hat{q}\)

For c), \(n = 64\), \(X = 16\). \[\hat{p} = \frac{16}{64} = 0.25\]. Then \(\hat{q} = 1 - \hat{p} = 1 - 0.25 = 0.75\).

Case D: Calculate \(\hat{p}\) and \(\hat{q}\)

For d), \(n = 200\), \(X = 175\). \[\hat{p} = \frac{175}{200} = 0.875\]. Then \(\hat{q} = 1 - \hat{p} = 1 - 0.875 = 0.125\).

Case E: Calculate \(\hat{p}\) and \(\hat{q}\)

For e), \(n = 148\), \(X = 16\). \[\hat{p} = \frac{16}{148} \approx 0.108\]. Then \(\hat{q} = 1 - \hat{p} \approx 1 - 0.108 = 0.892\).

Key concepts

Sample Proportion

The sample proportion is a way to express how many times a desired outcome, known as a "success," occurs within a set of trials. For instance, in a survey, if 20 people said "yes" out of 100 surveyed, the sample proportion is the count of "yes" responses divided by the total number surveyed. You can calculate it with the formula

• \( \hat{p} = \frac{X}{n} \)

Where \(X\) is the number of successful trials and \(n\) is the total number of trials. Here, "successful trials" simply refer to whatever condition or outcome you are measuring. It's important, though, not to confuse this with the probability of success, as the binomial proportion varies from sample to sample. It's an estimation of reality rather than the exact truth.

Successful Trials

A successful trial is when a specific desired outcome occurs during an individual trial in an experiment. For example, flipping a coin and getting heads can be a successful trial if heads were the desired outcome. To find out how successful an experiment was, we count the number of these successful trials (denoted as \(X\)). This count is a central element in calculating the sample proportion. The more successful trials you have, generally, the higher your sample proportion will be.
The proportion of successes, \(X\), over the total number of trials, \(n\), shows the relative occurrence of success, guiding decisions based on this estimation.

Binomial Setting

A binomial setting implies that there are two potential outcomes for each trial: success or failure. This setting makes calculations straightforward since each trial only needs to be evaluated in terms of whether it achieved the targeted result or not.
In this context, consider the following:

• Each trial (such as a poll or an experiment) is independent of others.
• There remains a constant probability of success throughout trials.
• For mathematical models and estimations, when these criteria are met, the results become much more standard and predictable through the estimation processes such as the sample proportion.

This environment of conducting tests and experiments underlies the exercise of computing \( \hat{p} \). It also supports understanding binomial distribution, which is foundational when working with binary outcomes.

Estimation of Proportions

Estimation of proportions is part of inferential statistics and serves as a crucial tool for data analysis. Through this process, we derive conclusions about population parameters based on sample statistics. We are often interested in proportions to understand viewpoints, behaviors, or binary outcomes in larger populations.

• \( \hat{p} \) serves as an estimate of the population proportion, offering a rough idea of the larger picture.
• It's important to note that estimates will have errors. To account for this, methods like confidence intervals and hypothesis tests are used.

Ultimately, with enough trials and proper data collection, these estimations help validate decisions and offer insight into trends and predictions.