Statistics Quiz with Solutions (5 Questions)

In statistics, the term **"population"** refers to the complete set of items or individuals that share one or more characteristics and are of interest for some question or experiment. A population can include people, objects, events, observations, or measurements. The defining feature of a population is that it encompasses all possible subjects that fit the criteria of the study or the question being asked, not just a subset.

For example:
- If you are studying the average height of adult men in a particular country, the population would include all adult men in that country.
- In a study examining the lifespan of a particular model of car, the population would be every car of that model.

Researchers often collect data from a **sample**, which is a subset of the population, to make inferences or predictions about the population because it's usually impractical or impossible to collect data from every member of the population. The goal is for the sample to be representative of the population, allowing conclusions drawn from the sample to be generalized back to the entire population.

To find the \(z\) value that satisfies \(P(Z \leq z) = 0.5636\) for a standard normal variable \(Z\), we look up the given probability in the standard normal distribution table or use a statistical software/tool that can invert the cumulative distribution function (CDF) of the standard normal distribution. This process is commonly referred to as finding the \(z\)-score or the percentile point.

The \(z\) value that satisfies the expression \(P(Z \leq z) = 0.5636\) for a standard normal variable \(Z\) is approximately \(0.16\).

To calculate the median of the sample heights, we first need to sort the heights in ascending order and then find the middle value (or the average of the two middle values if the sample size is even). Here's how it's done step-by-step with the given heights:

Given heights (in inches): {62.3, 68, 64, 66.1, 65.4, 61.8, 65.2, 68.6}

1. **Sort the heights in ascending order**:
- Sorted heights: {61.8, 62.3, 64, 65.2, 65.4, 66.1, 68, 68.6}

2. **Determine the sample size (n)**:
- The sample size (n) = 8 (even number)

3. **Find the middle position(s) since n is even**:
- Since n = 8, the middle positions are the 4th and 5th numbers in the sorted list.

4. **Identify the middle values**:
- The 4th number is 65.2, and the 5th number is 65.4.

5. **Calculate the median**:
- The median is the average of the 4th and 5th numbers: (65.2 + 65.4) / 2 = 65.3 inches.

Therefore, the median height of the sampled students is 65.3 inches. This value represents the midpoint of the height distribution in the sample, with half of the students having heights below this value and half having heights above it.

In the context of probability and statistics, when we discuss the **probability distribution** of a continuous random variable \(X\), we're focusing on how probabilities are allocated to various outcomes \(X\) can achieve within a continuous range. Unlike discrete random variables, which possess specific, countable outcomes, continuous random variables can assume any value within an interval, meaning their potential outcomes are uncountably infinite.

The **density function**, denoted as \(f(x)\), is crucial for defining the probability distribution of a continuous random variable. This function is characterized by several key features:

1. **Non-negativity**: \(f(x) \geq 0\) for all \(x\) within the domain of \(X\).
2. **Total area equals 1**: The integral of the density function \(f(x)\) over all possible values of \(X\) equals 1. This condition mirrors the concept that the probability of \(X\) assuming any value within its range is 100%.

To determine the probability that \(X\) lies within a certain interval \([a, b]\), we integrate the density function \(f(x)\) over this interval:

\[P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx\]

This calculation aggregates the "density" of \(X\) across the interval from \(a\) to \(b\), indicating the probability that \(X\) falls within that range. Given \(X\)'s continuity, the probability of \(X\) equalling any singular, precise value is \(0\). Probabilities are thus always computed over ranges of values.

In essence, the **probability distribution** of a continuous random variable, as outlined by its density function, offers a detailed map of how probabilities are distributed across different intervals of values that the variable can take on.

To find \(P(o)\), we use the fact that the sum of all probabilities in a sample space must equal 1. We are given:

- \(P(l) = 0.3\)
- \(P(m) = 0.3\)
- \(P(n) = 0.2\)

To find \(P(o)\), we subtract the sum of the probabilities of \(l\), \(m\), and \(n\) from 1:

\[P(o) = 1 - (P(l) + P(m) + P(n))\]

Substituting the given values:

\[P(o) = 1 - (0.3 + 0.3 + 0.2)\]
\[P(o) = 1 - 0.8\]
\[P(o) = 0.2\]

Thus, \(P(o) = 0.2\), meaning the probability of outcome \(o\) is 0.2.