Statistics Quiz with Solutions (10 questions)

The complement of a set \(B\) within a sample space \(S\) consists of all the elements in \(S\) that are not in \(B\). Given the sample space \(S = \{a, b, c, d, e\}\) and the set \(B = \{b, d\}\), the complement of \(B\) includes all elements in \(S\) except for those in \(B\).

Therefore, the complement of \(B\) is \(\{a, c, e\}\), since these are the elements in the sample space \(S\) that are not in \(B\). This means that option d is correct, illustrating the concept of set complement where you include all elements of the universal set (in this case, the sample space \(S\)) that are not part of the set in question (\(B\)).

In probability theory, an "event" refers to any subset of a sample space that can occur when a random experiment is performed. It is a collection of possible outcomes from the sample space. For example, in a dice-rolling experiment, the sample space is all possible outcomes \(\{1, 2, 3, 4, 5, 6\}\), and an event could be rolling an even number, represented as the subset \(\{2, 4, 6\}\). Therefore, an event is not just a random selection but a specific outcome or set of outcomes that we are interested in observing within the framework of probability and statistics. This makes option c, "event," the correct terminology for describing a subset of a sample space in the context of probability theory.

The mean (or expected value) of a binomial random variable is given by the formula:
\[ \mu = n \cdot p \]
where:
- \(n\) is the number of trials,
- \(p\) is the probability of success on each trial.

Given the parameters \(n = 10\) and \(p = 0.3\), we can calculate the mean as follows:
\[ \mu = 10 \cdot 0.3 = 3 \]

Therefore, the expected value (mean) of a binomial distribution with these parameters is 3. This means that, on average, we can expect 3 successes out of 10 trials if each trial has a probability of success of 0.3. This aligns with option b, making it the correct answer.

A random variable is a numerical quantity that results from a random experiment, where the outcome is uncertain. It assigns a numerical value to each outcome in the sample space of a stochastic (or random) process. The key characteristic of a random variable is its ability to take different values based on the inherent randomness of the process being observed. Unlike observable, numerical, expected, or experimental variables, a random variable specifically captures the concept of uncertainty and variability in outcomes, making it a fundamental concept in probability and statistics.

For large sample hypothesis tests concerning a single population proportion, the test statistic follows a standard normal distribution, especially as the sample size increases, making the approximation to the standard normal distribution more accurate due to the central limit theorem.

A statistic is a numerical value that summarizes or describes an aspect of a sample of data. It is derived from the sample data itself and is used to make inferences about the population from which the sample was drawn. Unlike a parameter, which describes a characteristic of the entire population, a statistic specifically refers to the characteristics of the sample. Examples of statistics include the sample mean, sample variance, and sample standard deviation, among others. Therefore, the term "statistic" best fits the description of a number that summarizes some aspect of the sample data.

The standard error (SE) of the sample proportion is calculated using the formula:
\[ SE = \sqrt{\frac{p(1-p)}{n}} \]
where \(p\) is the population proportion, and \(n\) is the sample size.

Given:
- \(p = 0.48\)
- \(n = 169\)

Plugging these values into the formula gives:
\[ SE = \sqrt{\frac{0.48(1-0.48)}{169}} \]

After calculating, the standard error (SE) of the sample proportion is approximately \(0.038\), which when rounded to two decimal places is \(0.04\).

The mean of the sample proportion, often denoted as \(\hat{p}\), for random samples is equal to the population proportion \(p\). This is because the expected value (mean) of the sample proportion is the same as the population proportion from which the samples are drawn.

Given that the population proportion \(p = 0.35\), regardless of the sample size (in this case, \(n = 900\)), the mean of the sample proportion \(\hat{p}\) will also be \(0.35\). This principle is rooted in the idea that, on average, the sample will accurately reflect the population characteristics if the sample is random and the size is sufficiently large.

Therefore, the correct answer is \(0.35\), which indicates that the mean of the sample proportion obtained from random samples of size 900 is expected to be equal to the population proportion of \(0.35\).

The probability \(P(-2.05 < Z < 1.96)\) for a standard normal variable \(Z\) represents the area under the standard normal distribution curve between the z-scores of \(-2.05\) and \(1.96\). The standard normal distribution is symmetric around zero, and the areas (probabilities) associated with specific z-score ranges can be found using the standard normal distribution table or a calculator.

The z-scores of \(-2.05\) and \(1.96\) roughly correspond to the 2.5th percentile and the 97.5th percentile of the distribution, respectively. The total area (probability) between these two z-scores encompasses most of the distribution, leaving about 2.5% in the tails outside of this range. Therefore, the combined area under the curve between \(-2.05\) and \(1.96\) is approximately 95% of the total area, which corresponds to a probability of 0.9548.

This calculation is based on the properties of the standard normal distribution and is a common way to approximate probabilities for normally distributed data. The exact values may vary slightly based on the precision of the tables or computational tools used, but the option closest to the theoretical value is the correct answer.

The expression \(P(Z \leq z) = 0.7734\) refers to the probability that the standard normal variable \(Z\) is less than or equal to a certain z-value, which in this case corresponds to a probability of 0.7734. This probability is associated with the area under the standard normal distribution curve to the left of the specified z-value.

To find the correct z-value, one typically refers to a z-table (standard normal distribution table) or uses a statistical calculator that can invert this probability into a z-score. The value of 0.7734 suggests a z-score that is to the right of the mean (since it's more than 0.5, indicating more than half of the area under the curve is to the left of this point).

A z-score of 0.75 is approximately where the cumulative probability from the left up to this z-score matches 0.7734 in the standard normal distribution. Therefore, the correct z-value that satisfies \(P(Z \leq z) = 0.7734\) is 0.75, indicating that about 77.34% of the distribution falls to the left of this point on the z-distribution curve. This methodology is based on the properties of the standard normal distribution, where probabilities are mapped to z-scores that represent standard deviations from the mean.