Statistics and Probability

A sampling method where the population is easily accessible to the researcher is called:

What is the range of the following data: -5, -2, 0, 3, 7?

Which of the following values of $r$ indicates a perfect positive correlation?

If the mean of a set of 4 numbers is 8, what is the sum of the numbers?

A set of data has a standard deviation of 0. What does this imply about the data values?

The mean age of a group of 10 people is 24. If a person aged 35 leaves the group, what is the new mean age of the remaining 9 people?

What is the primary advantage of using a stratified sample over a simple random sample?

The mean of four numbers $a, b, c, d$ is 10. If the mean of $a, b, c$ is 8, find the value of $d$.

A set of 100 data points has a mean of 10. A second set of 50 data points has a mean of $m$. The combined mean is 12. Find $m$.

The equation of the regression line of $y$ on $x$ is $y = 2x + 3$. If $x$ is measured in cm and $y$ is measured in mm, and we change $x$ to be measured in mm, what is the new slope?

Calculate the mean of the first five prime numbers.

A data set has a range of 20. If every value is divided by 2, what is the new range?

In a data set of 100 values, which value represents the 75th percentile?

If the mean of $a, b, c$ is 10, what is the mean of $2a, 2b, 2c$?

Given $n=10$, $\bar{x}=15$, and $\sum x^2 = 2350$. Calculate the variance.

A data set has a variance of $s^2$. If every value is divided by 2, what is the new variance?

The median of 5 numbers $x, x+1, x+3, x+4, x+10$ is 12. Find $x$.

A data set with $n$ observations has a mean of $\mu$ and a variance of $\sigma^2$. Two additional observations, $x_{n+1} = \mu + k$ and $x_{n+2} = \mu - k$, are added to the set. Given that the variance of the new set of $n+2$ observations is $\frac{3}{2}\sigma^2$, find the expression for $k^2$ in terms of $n$ and $\sigma^2$.

The variance of a set of data is 16. If each value is divided by 2 and then 3 is added, find the new standard deviation.

Given that the mean of $x_1, x_2, \dots, x_{10}$ is 15, find $\sum_{i=1}^{10} (2x_i - 5)$.

Calculate the mid-interval value for $0.5 \le x < 0.9$.

Which class is the median class for $n=20$ if frequencies are 5, 4, 3, 8?

If the class width is 20 and the lower boundary is 100, what is the upper boundary?

Find the estimated mean for: Class 1 (midpoint 5, freq 4), Class 2 (midpoint 15, freq 6).

A dataset is grouped into classes $0 \le x < 10$ and $10 \le x < 20$. If the frequencies are 10 and 30 respectively, what is the estimated median?

The standard deviation of a grouped data set is 5. If every value in the data set is multiplied by 3, what is the new variance?

In a histogram, if the class width is doubled and the frequency is halved, how does the height of the bar change to keep the area proportional to frequency?

The variance of a grouped frequency distribution is $S^2$. If the class intervals are $[0, 10), [10, 20), ...$ and we change them to $[0, 5), [5, 10), [10, 15), ...$ while maintaining the same underlying data distribution, how would the new estimated variance $S'^2$ compare to $S^2$?

A frequency table has classes $[0, 10), [10, 20), [20, 30)$. Frequencies are $2, 3, 5$. If the frequency of the first class is doubled, by how much does the estimated mean decrease?

The estimated mean of a grouped frequency table is $M$. If the midpoint of every class is increased by $10\%$ and every frequency is decreased by $10\%$, what is the new estimated mean?

Which of the following best describes the correlation between the age of a second-hand car and its market price?

A data set has $x$-values $2, 4, 6, 8$. If you use the regression line to estimate $y$ for $x = 5$, you are performing:

If the correlation coefficient is $r = 0.99$, the linear relationship is described as:

Given the regression line $y = 3x - 5$, find the value of $y$ when $x = 0$.

The line of best fit for a data set is $y = -1.2x + 15$. If the range of $x$ values is $2 \le x \le 10$, which of the following is an interpolation?

A dataset with $n=8$ has $\bar{x} = 10$ and $\bar{y} = 20$. If a point $(10, 20)$ is removed, what is the new mean point?

What is the purpose of calculating the Pearson product-moment correlation coefficient?

The regression line $y = ax + b$ is calculated. If the mean of $x$ is doubled and the mean of $y$ is tripled, but the gradient $a$ remains the same, the new $y$-intercept $b'$ is given by:

A set of data has the mean point $(k, 2k)$. The regression line of $y$ on $x$ is $y = 1.5x + 5$. Calculate the value of $k$.

If $r^2 = 0.49$ and the gradient of the regression line is negative, what is the value of $r$?

If $n(A) = 7$ and $n(B) = 3$, what is the maximum possible value of $n(A \cup B)$?

If $n(A) = 7$ and $n(B) = 3$, what is the minimum possible value of $n(A \cup B)$?

Which operation is commutative?

If $A = \emptyset$, what is $n(A)$?

In a set notation context, which symbol represents 'is a subset of'?

Given $P(A') = 0.7$ and $P(B') = 0.8$. If $A$ and $B$ are independent, find $P(A \cup B)$.

A set $A$ has 3 elements. How many non-empty subsets does $A$ have?

Given $P(A) = p$, $P(B) = p$, and $P(A \cap B) = p^2$. Express $P(A' \cap B')$ in terms of $p$.

If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cup B) = 0.7$, find $P(A' \cap B)$.

Events $E$ and $F$ satisfy $P(E | F) = P(E)$. This implies:

If $X \sim B(3, 0.5)$, calculate the probability $P(X=1)$.

A bag has 4 red and 6 green balls. Two balls are drawn without replacement. What is the probability the first ball is red?

For a discrete random variable $X$, what is the value of $\sum P(X=x)$ over all possible values of $x$?

In a Venn diagram with two overlapping circles $A$ and $B$, the region $A \cap B'$ represents:

If $P(A) = 0.5$ and $P(B) = 0.5$, and $P(A \cup B) = 0.8$, find $P(A \cap B)$.

Find the probability of rolling a sum of 12 with two fair six-sided dice.

Given $P(A) = 0.6, P(B) = 0.3$. If $P(A \cap B) = 0.18$, are $A$ and $B$ mutually exclusive?

A random variable $X$ has $E(X) = \mu$. Find $E(X - \mu)$.

Two events $A$ and $B$ are such that $P(A) = 0.3, P(B) = 0.4$. If $P(A|B) = 0.5$, find $P(A \cup B)$.

If $X \sim B(n, p)$, and $P(X=k) = P(X=n-k)$, what must be the value of $p$?

If $P(A|B) = 0.25$ and $P(B) = 0.4$, find $P(A \cap B)$.

If $P(A) = 0.4$ and $P(B) = 0.5$ and $P(A \cap B) = 0.2$, find $P(A \cup B)$.

In a sample space $S$, $P(A) = 0.4$ and $P(B) = 0.2$. If $A$ and $B$ are independent, find $P(A' \cap B')$.

Given $P(A \cap B) = 0.3$ and $P(A) = 0.6$, find $P(B|A)$.

Given $P(A) = 0.6, P(B) = 0.4$, and $P(A \cap B) = 0.24$, calculate $P(A'|B')$.

If $P(A) = 0.4$ and $P(B) = 0.5$ and the events are independent, find the probability that exactly one of the events occurs, $P(A \Delta B)$.

If $P(A) = 0.3$ and $P(A|B) = 0.3$, then events $A$ and $B$ must be:

Two events $X$ and $Y$ are independent such that $P(X) = 0.5$ and $P(X \cup Y) = 0.7$. Find $P(Y)$.

In a survey, $70\%$ of people like product A, $60\%$ like product B, and $20\%$ like neither. Find the probability a person likes product A given they like product B.

If $P(A) = 0.5$ and $P(B) = 0.2$, and $A$ and $B$ are independent, find $P(A' | B)$.

A box has $4$ red and $1$ white ball. If $2$ balls are drawn without replacement, what is the probability that the first is red and the second is white?

In a tree diagram, $P(A) = 0.6$. Given $A$, $P(B|A) = 0.3$. Given $A'$, $P(B|A') = 0.3$. Are $A$ and $B$ independent based on this tree?

A coin is flipped and then a $6$-sided die is rolled. Find the probability of getting 'Tails' and a '4'.

The probability of event $E$ is $0.4$. The probability of event $F$ given $E$ is $0.5$. Find $P(E \cap F')$.

On any given day, the probability that Dan cycles to work is $0.7$. If he cycles, the probability he buys lunch is $0.4$. If he doesn't cycle, the probability he buys lunch is $0.9$. Given that he bought lunch today, find the probability he cycled.

A student has two bags of coins. Bag 1 has 4 gold and 6 silver coins. Bag 2 has 7 gold and 3 silver coins. They pick a bag and draw two coins without replacement. If Bag 1 is picked with probability $0.4$, find the probability of drawing two gold coins.

In a soccer match, the probability that Team A scores first is $0.6$. If they score first, they win the match with probability $0.8$. If they do not score first, they win with probability $0.3$. Find the probability that Team A wins the match.

A box contains $n$ red balls and 5 blue balls. If two balls are drawn without replacement and the probability of drawing two blue balls is $\frac{2}{9}$, find $n$.

A diagnostic test has a probability 0.95 of giving a positive result if the patient has the disease, and 0.04 of giving a positive result if the patient does not. If 2% of the population has the disease, find the probability a patient has the disease given they tested negative.

A bag contains 4 red and 2 blue marbles. Three marbles are drawn without replacement. What is the probability that the first and third are red and the second is blue?

In a distribution where $P(X=x) = \frac{1}{3}$ for $x=k, 2k, 3k$, and $E(X) = 6$, find $k$.

Given $P(X=x) = \frac{x}{6}$ for $x=1, 2, 3$. Find $E(X)$.

A variable $X$ has $P(X=1)=0.1, P(X=2)=0.2, P(X=3)=0.3, P(X=4)=0.4$. Find $P(1 < X < 4)$.

If $E(X) = 7$, find $E(2X - 14)$.

A random variable $X$ has $E(X)=5$. Find $E(2X+3)$.

In a distribution $B(n, p)$, if $n=10$ and $P(X=0) = (0.9)^{10}$, find $E(X)$.

For a discrete distribution where $P(X=1)=a$ and $P(X=2)=a^2$ and $P(X=3)=a$, where $a>0$. Find the value of $a$.

A random variable $X$ has $E(X) = 2$ and $E(X^2) = 8$. Find $Var(3X + 1)$.

In a binomial distribution $B(n, p)$, if $n=5$ and $P(X=1) = 2 P(X=0)$, find $p$.

A discrete random variable $X$ has $P(X=x) = p(1-p)^{x-1}$ for $x=1, 2, 3, ...$. If $p=0.5$, find $P(X \leq 3)$.

For $X \sim B(100, 0.01)$, find the mean.

If $n = 6$, how many possible values can the random variable $X$ take?

Which of these pairs of parameters uniquely defines a binomial distribution?

For $X \sim B(5, 0.4)$, find $P(X \ge 1)$.

For $X \sim B(n, p)$, the mean is 20 and the standard deviation is 4. Find the value of $p$.

Let $X \sim B(10, 0.3)$. Find $P(X = 1) + P(X = 0)$, giving your answer in terms of powers.

A seed has a $70\%$ chance of germinating. If 5 seeds are planted, find the probability that exactly 4 germinate.

Find the smallest $n$ such that $X \sim B(n, 0.5)$ satisfies $P(X \ge 1) \ge 0.999$.

In a binomial distribution $X \sim B(n, p)$, the mean is 5 and the variance is 4. Find $n$.

Let $X \sim B(n, p)$. If $n=5$ and $P(X=2) = 2 P(X=3)$, find $p$.

The area under the normal curve to the left of the mean is:

If $\mu = 60$ and $x = 54$ corresponds to a $z$-score of $-1.2$, what is the standard deviation?

For $Z \sim N(0, 1)$, what is $P(Z > 0)$?

If $X \sim N(50, 100)$, find the $z$-score for $x = 40$.

In a normal distribution, if the interquartile range (IQR) is 20, find the approximate value of $\sigma$. (Hint: $Q_3$ is at $z \approx 0.674$)

Find $P(Z < -0.5)$ if $Z \sim N(0, 1)$.

Which property is NOT true for a normal distribution curve?

The probability that a person is taller than 180 cm is 0.2. Heights are normally distributed with a standard deviation of 5 cm. Find the mean height of the population.

If $X \sim N(\mu, \sigma^2)$, what is the ratio of $P(\mu - 2\sigma < X < \mu + 2\sigma)$ to $P(\mu - \sigma < X < \mu + \sigma)$?

The length of a commute follows $N(\mu, \sigma^2)$. 5% of commutes take longer than 60 minutes and 2% take less than 20 minutes. Find the standard deviation $\sigma$.

If a continuous random variable $X$ has a constant PDF $f(x) = c$ over $[a, b]$, the distribution is called:

In a standard normal distribution, what is the area between $z = -1$ and $z = 1$ approximately?

If $F(x)$ is the CDF of $X$, what is the derivative $\frac{d}{dx}F(x)$?

If $f(x) = 6x(1-x)$ for $0 \le x \le 1$, calculate the mode.

A continuous random variable $X$ has the probability density function $f(x)$ defined by $f(x) = \frac{1}{2} \cos \left( \frac{x}{2} \right)$ for $0 \le x \le \pi$. Find the median of $X$.

If $X \sim N(0, 1)$, find $P(X > 1.96)$.