Fundamentals Of Statistical And Mathematical Frameworks For Ai Practice Tests

Question 1

A data scientist is cleaning a dataset where the number of missing values $M$ in a feature follows the relationship $M = 0.05N$, where $N$ is the total number of observations. If $N = 2000$, calculate the number of valid observations remaining after removing all missing entries.

Accepted Answer

1900

Answer

100

Answer

2000

Answer

2100

Question 2

In a disaster management scenario, the probability $P$ of an infrastructure failure is modeled by $P(t) = 1 - e^{-\lambda t}$, where $\lambda = 0.02$ per hour. Calculate the probability of failure at $t = 50$ hours. Provide the result as a decimal rounded to four places.

Accepted Answer

0.6321

Question 3

A city health department tracks vaccination rates $V$ using the function $V(x) = \frac{1}{1 + e^{-k(x-x_0)}}$, where $x$ represents the population density index. If $k=0.5$ and $x_0=2$, determine the vaccination rate when $x=4$.

Accepted Answer

0.7311

Answer

0.2689

Answer

0.5000

Answer

0.8808

Question 4

If a weather forecasting model compares current temperature $T_c$ with historical mean $T_h$ and finds the difference $\Delta T = |T_c - T_h| > 3\sigma$, where $\sigma$ is the standard deviation, the observation is statistically considered an outlier.

Accepted Answer

True

Answer

False

Question 5

Which of the following scenarios represent valid applications of the statistical process (Collection, Exploration, Analysis, Decision) in AI-driven domains?

Answer

Using historical weather data to predict future storm paths

Answer

Analyzing COVID-19 infection trends to determine Olympic event scheduling

Answer

Identifying high-risk vaccination zones based on population density data

Answer

Manually guessing infrastructure repair needs without data input

Question 6

A sports analyst calculates the moving average of athlete performance scores $S_n$ using the formula $S_n = \frac{1}{k} \sum_{i=n-k+1}^{n} x_i$. If $k=3$ and the last three scores are $x_1=80, x_2=85, x_3=90$, calculate the moving average $S_3$.

Accepted Answer

85

Question 7

In a disease prediction model, the risk score $R$ is calculated as $R = \sum_{i=1}^{n} w_i x_i$. If weights $w = [0.4, 0.6]$ and indicators $x = [10, 5]$, calculate the total risk score $R$.

Accepted Answer

7

Answer

5

Answer

6

Answer

8

Question 8

An insurance company models the probability of a claim occurring for a specific policyholder as $P(A) = 0.04$. If the company insures $1000$ independent policyholders, calculate the expected number of claims $E[X]$ using the formula $E[X] = n \cdot p$.

Accepted Answer

40

Answer

4

Answer

400

Answer

0.4

Question 9

If a meteorologist states there is a $0.85$ probability of rain, this event is classified as 'Likely' because $P(E) > 0.5$.

Accepted Answer

True

Answer

False

Question 10

A fair six-sided die is rolled. Determine the probability $P(E)$ of rolling a number greater than $6$. Express your answer as a decimal.

Accepted Answer

0

Question 11

In a medical study, a treatment has a success rate of $P(S) = 0.75$. If two patients are treated independently, what is the probability that both treatments are successful, calculated as $P(S_1 \cap S_2) = P(S_1) \cdot P(S_2)$?

Accepted Answer

0.5625

Answer

0.75

Answer

1.5

Answer

0.5

Question 12

Which of the following scenarios represent an 'Equal Probability' event? Select all that apply.

Answer

Tossing a fair coin and getting Heads or Tails

Answer

Rolling a fair four-sided die and getting any specific face

Answer

Predicting the weather for tomorrow

Answer

Drawing a card from a deck and getting a Joker

Question 13

An insurance firm calculates the premium $R$ for a risk $L$ using the formula $R = P(L) \cdot L + C$, where $P(L) = 0.02$, $L = €50,000$, and administrative cost $C = €200$. Calculate the premium $R$.

Accepted Answer

€1200

Question 14

An AI model uses a weight matrix $W = \begin{pmatrix} 2 & 1 \ 0 & 3 \end{pmatrix}$ to transform an input vector $v = \begin{pmatrix} 1 \ 2 \end{pmatrix}$. Calculate the resulting vector $Wv$.

Accepted Answer

$\begin{pmatrix} 4 \ 6 \end{pmatrix}$

Answer

$\begin{pmatrix} 2 \ 6 \end{pmatrix}$

Answer

$\begin{pmatrix} 3 \ 3 \end{pmatrix}$

Answer

$\begin{pmatrix} 1 \ 5 \end{pmatrix}$

Question 15

To optimize a model, an AI calculates the gradient of a loss function $f(x) = 3x^2 + 5x$. Determine the derivative $f'(x)$ to find the direction of steepest ascent.

Accepted Answer

$6x + 5$

Answer

$3x + 5$

Answer

$x^2 + 5$

Answer

$6x^2 + 5$

Question 16

Identify which mathematical branches are primarily responsible for the following AI tasks: (Select all that apply)

Answer

Statistics: Analyzing data distribution

Answer

Linear Algebra: Processing multi-dimensional data arrays

Answer

Calculus: Storing permanent database records

Answer

Probability: Calculating the exact physical weight of hardware

Question 17

A spam filter uses Bayes' Theorem to determine the probability that an email is spam given the presence of the word 'win'. If $P(S|W) = \frac{P(W|S)P(S)}{P(W)}$, and $P(W|S) = 0.8$, $P(S) = 0.1$, and $P(W) = 0.2$, calculate $P(S|W)$.

Accepted Answer

0.4

Question 18

In decision-making under uncertainty, AI models prefer rigid, deterministic rules over probabilistic estimates to handle noisy real-world data.

Accepted Answer

False

Answer

True

Question 19

An autonomous vehicle calculates the probability of a pedestrian crossing as $P(C) = 0.75$. What is the probability that the pedestrian does NOT cross, represented as $P(C^c)$?

Accepted Answer

0.25

Question 20

If a voice assistant has a 90% confidence level for command A and a 10% confidence level for command B, what is the mathematical basis for choosing command A?

Accepted Answer

Maximum Likelihood Estimation

Answer

Linear Regression

Answer

Matrix Inversion

Answer

Integral Calculus

Fundamentals Of Statistical And Mathematical Frameworks For Ai Practice Tests

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Fundamentals of Statistical and Mathematical Frameworks for AI

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