Statistics Quiz with Solutions (6 Questions)

To find the probability that a random variable \( X \), uniformly distributed on the interval [0, 1], assumes a value greater than 0.90, we can use the properties of the uniform distribution.

The probability density function \( f(x) \) of a uniform distribution on the interval [a, b] (in this case, [0, 1]) is defined as:
\[
f(x) = \frac{1}{b - a}
\]
For the interval [0, 1], this simplifies to:
\[
f(x) = 1 \text{ for } x \in [0, 1]
\]

The probability that \( X \) is greater than a particular value \( c \) within [0, 1] is calculated as the length of the interval from \( c \) to 1. Specifically, for \( X > 0.90 \):
\[
P(X > 0.90) = 1 - 0.90 = 0.10
\]

Hence, the probability that \( X \) assumes a value greater than 0.90 is 0.10, or 10%.

Calculating \( \frac{4}{6} \times 100 \) yields approximately 66.67%. Thus, if we're directly using the position in the dataset where the total number of elements equals 6 and 2.9 is the fourth element, 2.9 is at the 66.67th percentile. This method is straightforward, counting the index directly against the total number of elements.

When two balanced dice are rolled, the total number of possible outcomes is \(6 \times 6 = 36\), since each die has 6 faces.

To find the probability of not rolling doubles, we first calculate the probability of rolling doubles. Rolling doubles happens when both dice show the same number, which can occur in the following combinations: \( (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \). There are 6 such outcomes.

Thus, the probability of rolling doubles is:
\[
\frac{6}{36} = \frac{1}{6}
\]

To find the probability of not rolling doubles, we subtract the probability of rolling doubles from 1:
\[
1 - \frac{1}{6} = \frac{5}{6}
\]

Hence, the probability of not rolling doubles is \(\frac{5}{6}\).

As the sample size of a random sample increases, the standard deviation of the sample mean **decreases**. This concept is reflected in the formula for the standard error of the mean, which is the standard deviation of the sample mean:

\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\]

where \( \sigma \) is the population standard deviation and \( n \) is the sample size. As \( n \) increases, \( \sqrt{n} \) increases, making the denominator larger and thus reducing the standard error.

A **sample** is any subset or subcollection of the population.

A **measurement** is a number or attribute computed for each member of a population or of a sample.