Question

Find each \(X,\) given \(\hat{p}\) a. \(\hat{p}=0.60, n=240\) b. \(\hat{p}=0.20, n=320\) c. \(\hat{p}=0.60, n=520\) d. \(\hat{p}=0.80, n=50\) e. \(\hat{p}=0.35, n=200\)

Short answer

a. 144, b. 64, c. 312, d. 40, e. 70

Step-by-step solution

Understand the Problem

The problem provides the sample proportion \( \hat{p} \) and sample size \( n \) for a set of examples. The goal is to calculate the expected number, \( X \), of successful outcomes in the sample, which can be found using the formula \( X = \hat{p} \times n \).

Calculate X for Part a

For the given \( \hat{p} = 0.60 \) and \( n = 240 \), calculate \( X \) using \( X = \hat{p} \times n \). Therefore, \( X = 0.60 \times 240 = 144 \).

Calculate X for Part b

For \( \hat{p} = 0.20 \) and \( n = 320 \), compute \( X \) as \( X = \hat{p} \times n = 0.20 \times 320 = 64 \).

Calculate X for Part c

Given \( \hat{p} = 0.60 \) and \( n = 520 \), use the formula to find \( X = \hat{p} \times n = 0.60 \times 520 = 312 \).

Calculate X for Part d

For \( \hat{p} = 0.80 \) and \( n = 50 \), calculate \( X = \hat{p} \times n = 0.80 \times 50 = 40 \).

Calculate X for Part e

With \( \hat{p} = 0.35 \) and \( n = 200 \), find \( X \) using \( X = \hat{p} \times n = 0.35 \times 200 = 70 \).

Key concepts

Sample Proportion

In statistics, understanding the concept of **sample proportion** is crucial. The sample proportion, typically denoted as \( \hat{p} \), represents the fraction of successes in a sample. For instance, in a survey where 150 out of 300 respondents prefer chocolate ice cream, the sample proportion \( \hat{p} \) is 0.50. It serves as an estimate of the true proportion (or probability) of success in the entire population.

• The sample proportion enables statisticians to infer population characteristics.
• It is calculated as \( \hat{p} = \frac{x}{n} \), where \( x \) represents the number of successes in the sample and \( n \) is the sample size.
• This metric allows researchers to make predictions and test hypotheses.

Grasping sample proportions helps in solving problems related to population dynamics and surveys.

Sample Size

The **sample size**, indicated by \( n \), is the total number of observations or elements in a sample. It is a pivotal factor in research and impacts the reliability of the results. A larger sample size generally increases the accuracy of statistical analyses and reduces sampling error.

• Larger samples give more reliable estimates of population parameters.
• However, larger samples might require more resources and time to collect.
• The selection of an adequate sample size depends on the study's objectives and resources.

For example, when calculating how many people in a city prefer electric cars, the researcher must decide on an appropriate sample size to confidently extrapolate findings to the entire population.

Probability

**Probability** is a fundamental concept in statistics. It quantifies the likelihood of an event occurring, ranging from 0 (impossibility) to 1 (certainty). In the context of sample proportions, probability represents the chance of a chosen sample exhibiting certain traits.

• Probability helps in predicting future events based on existing data.
• It enables statisticians to make informed decisions with limited information.
• Sample proportion is a type of probability, detailing the likelihood of drawing a specific observation from a population.

For a deeper understanding, consider flipping a fair coin; the probability of getting heads is 0.5. Similarly, if 60% of a class votes for a particular candidate, the probability of randomly choosing a student who supports that candidate is 0.60.

Statistical Formulas

Statistical analysis relies heavily on various **formulas** that simplify complex calculations and provide insights into data trends. One such crucial formula involves finding the expected count of successes in a sample: \( X = \hat{p} \times n \).

• This formula calculates the expected number of successes given a sample proportion and size.
• Statistical formulas can be used across various fields to describe and predict outcomes.
• They offer a structured way of resolving problems and drawing conclusions from data.

Utilizing these formulas accurately can help in making sense of real-world data. For example, if \( \hat{p}=0.60 \) and \( n=100 \), then the expected number of successful outcomes \( X \) would be 60, signifying that 60% of the sample exhibits a particular trait. Understanding these equations is vital for translating statistical jargon into actionable insights.