Statistics and Probability

A survey asks respondents to rank their satisfaction from 1 (lowest) to 5 (highest). This data is:

In the context of data collection, what does 'bias' refer to?

When data is collected through an experiment where the researcher controls variables, the data is:

A dataset lists the colors of 50 cars: 20 Red, 15 Blue, 10 Silver, 5 White. What is the relative frequency of Red cars?

A researcher decides to investigate the relationship between age and screen time by asking their own family members. This is an example of:

In systematic sampling from a population of 800 people with a sample size of 40, if the first person chosen is at position 7, what is the position of the second person chosen?

Which of the following is a discrete variable?

A teacher wants to select a sample of 5 students from a class of 25. She lists the students alphabetically and chooses students at positions 5, 10, 15, 20, and 25. This is:

Which of these variables is qualitative ordinal?

A population $N=250$ is divided into strata $A$ ($N_A = 100$) and $B$ ($N_B = 150$). If a stratified sample contains 20 people from stratum $A$, how many are from stratum $B$?

A population of size $N=500$ is split into two strata of $N_1=200$ and $N_2=300$. A stratified sample of $n=50$ is taken. How many more people are taken from the second stratum than the first?

A survey asks the first 50 people who visit a website for their opinion. This type of sampling is:

A music festival has 5000 attendees. A researcher wants to sample 100 of them. If systematic sampling is used, what is the value of $k$?

In a study on the effect of a new fertilizer, a farmer divides his field into 4 quadrants and chooses one quadrant at random to test. This is an example of:

A survey was sent to 500 randomly selected customers of an online store. Only 50 customers replied. This low response rate likely leads to:

A stratified sample of 120 people is taken from a town of 6000. If 45 of those sampled are over 60 years old, estimate the total number of people over 60 in the town.

A researcher wants to compare the heights of students in two different schools. He takes a sample of 30 students from each school by picking the first 30 students who arrive at the school gate. What is the main problem with this approach?

A university has 5000 students. A researcher wants a sample of 100. They decide to pick student number 50 and then every 50th student after that. What is the number of the last student in the sample?

Using a systematic sample with $k=25$ and start $a=10$, what is the formula for the $n$-th term $x_n$?

A researcher takes a stratified sample of 150 people from a town of 15,000. Stratum A has 3,000 people and Stratum B has 12,000. How many more people are sampled from Stratum B than Stratum A?

The total area under a histogram where the vertical axis is 'Relative Frequency Density' (where total frequency is expressed as $1$) is always:

On a cumulative frequency curve, which percentile corresponds to the lower quartile?

If you are given a cumulative frequency table, how do you find the frequency of a single class interval?

What is the median of a dataset if its box plot has a minimum of $5$, $Q_1$ of $12$, $Q_3$ of $22$, and a maximum of $30$, and is perfectly symmetric?

A cumulative frequency graph is used to estimate the number of people who weigh less than $70$ kg. If the point on the curve is $(70, 145)$ and the total number of people is $180$, what fraction of people weigh more than $70$ kg?

A histogram has classes $0-5$, $5-15$, and $15-20$ with frequencies $10, 30, 10$ respectively. Which class(es) have the highest frequency density?

If a box plot shows that $Q_3 - \text{Median} > \text{Median} - Q_1$, the data is said to be:

A dataset $X$ contains 7 values: $\{10, 12, 13, 15, 17, 18, 25\}$. Let $x$ be the smallest integer that would be classified as an upper outlier for this dataset using the $1.5 \times IQR$ rule. If a second dataset $Y$ is created using the transformation $Y_i = 3X_i - 10$, what is the value corresponding to $x$ in dataset $Y$?

A set of continuous data is grouped into three classes: $0 \le x < 10$ (frequency 5), $10 \le x < 20$ (frequency $k$), and $20 \le x < 30$ (frequency 11). The median of the data, calculated by linear interpolation, is 18. Determine the 75th percentile ($Q_3$) for this dataset.

A dataset is represented by a histogram with two adjacent classes: $0 \le x < 20$ with frequency density $d_1$, and $20 \le x < 50$ with frequency density $d_2$. If the mean of the data (calculated using class midpoints) is 28, find the ratio $d_1 : d_2$.

Given the set $\{10, 20, 30, 40, 50, 60, 70\}$, calculate the $IQR$.

The difference between the third quartile ($Q_3$) and the first quartile ($Q_1$) is known as the:

If the mean of a dataset is $50$ and we add $5$ to every value, what is the new mean?

Find $Q_1$ for the dataset: $10, 11, 12, 13, 14, 15, 16$.

The interquartile range ($IQR$) of a data set is $15$. If every value is multiplied by $2$ and then $10$ is subtracted, what is the new $IQR$?

A data set has $n=20$ values with $\sum x = 160$. If a new value $x=34$ is added, what is the new mean?

Which of the following measures of central tendency is the most sensitive to extreme outliers?

Consider the set of integers $\{1, 2, 3, ..., n\}$. For what value of $n$ is the variance equal to 10?

The standard deviation of a set of 50 observations is 6.5. If another set of 50 observations has a standard deviation of 6.5 and both sets have the same mean, find the standard deviation of the combined set of 100 observations.

A data set of 5 numbers has a mean of 6 and a variance of 2. If the numbers are $4, 7, 5, 8,$ and $x$, find $x$.

Calculate the population standard deviation of the numbers 5, 5, 15, 15 using your GDC.

What is the interquartile range of the data: 10, 10, 10, 20, 20, 20?

In a box plot, the whiskers represent the distance from the quartiles to the maximum and minimum values (excluding outliers). If the min is 2, $Q_1$ is 5, $Q_3$ is 12, and the max is 20, what is the range?

Find the range of the following dataset: 102, 105, 101, 109, 104.

Find the range of the following frequency distribution:

Value: 10, 20, 30, 40 Frequency: 5, 2, 8, 1

A set of data has a standard deviation of 4. If all data values are tripled, what is the new variance?

A scientist measures the mass of several insects. The standard deviation is $0.12 \text{ g}$. If the scientist accidentally used a container weighing $0.5 \text{ g}$ for all insects and forgot to subtract it, how does the corrected standard deviation compare to the original?

A set of $5$ numbers has a mean of $8$ and a variance of $4$. If the numbers are doubled, what is the new sum of squares $\\sum y_i^2$?

If the IQR of a data set is $0$, what can be said about the middle $50\\%$ of the data?

The standard deviation of $n$ values is $s$. If each value is squared, which of the following is true about the new standard deviation?

The weights of four people are 60 kg, 62 kg, 64 kg, and 66 kg. Calculate the population standard deviation to two decimal places.

In a distribution where all data points are the same, the standard deviation is:

A student finds that the variance of their data is 1.44. What is the standard deviation?

If the standard deviation of $x$ is 5, what is the standard deviation of $2x + 10$?

For the data set: $5, 10, 15, 20, 25$, the mean is $15$. Calculate the sum of the squared deviations from the mean ($\sum (x_i - \bar{x})^2$).

A coach is comparing two swimmers. Swimmer 1 has a mean lap time of $60$ seconds with a standard deviation of $1.5$ seconds. Swimmer 2 has a mean lap time of $62$ seconds with a standard deviation of $0.5$ seconds. Which swimmer is more 'reliable' in terms of consistent lap times?

The variance of a sample is $s^2$. If every value in the sample is increased by a constant $c$, the new variance is:

A sample of size 1 has a mean of 10. What is the standard deviation of this sample?

If the standard deviation of $X$ is $\sigma_X$, and $Y = aX + b$, what is the relationship between the variance of $Y$ and the variance of $X$?

For a set of data, the mean is 50 and the standard deviation is 8. A value $x$ is added to the set such that the new mean is still 50. What must be the value of $x$?

What is the primary purpose of finding the line of best fit?

In a scatter diagram, if the points cluster tightly around a straight line, the magnitude of $r$ is:

What does a correlation coefficient of $r = 0.7$ suggest about the relationship between two variables?

If the correlation coefficient $r$ is positive, the line of best fit must have a:

A retail manager analyzes the relationship between the daily average temperature ($x$ in $^\circ\text{C}$) and the number of units of a specific cold beverage sold ($y$). The following statistics were calculated for the data set:

• Pearson's product-moment correlation coefficient: $r = 0.72$
• Standard deviation of temperature: $S_x = 2.5$
• Standard deviation of units sold: $S_y = 15$
• Mean temperature: $\bar{x} = 6$
• Mean units sold: $\bar{y} = 84$

Calculate the $y$-intercept of the least-squares regression line of $y$ on $x$.

A study is conducted to determine the relationship between the number of hours students spend on a digital learning platform ($x$) and their final exam grade ($y$ percentage). The mean time spent on the platform is $\bar{x} = 6.5$ hours and the mean exam grade is $\bar{y} = 72\%$. The least-squares regression line of $y$ on $x$ is known to pass through the point $(2, 45)$. Calculate the predicted exam grade for a student who spent $9$ hours on the platform.

A bivariate data set $(x, y)$ has $10$ points. If $\sum x = 40$ and $\sum y = 60$, and the regression line is $y = mx + 2$, what is the value of $m$?

A real estate agent uses a regression model to predict house prices $y$ (in thousands of dollars) based on the area $x$ (in $m^2$). The regression line is $y = 4x + 120$. A specific house has an area of $200$ $m^2$ and was sold for $935$ thousand dollars. Calculate the squared residual for this data point.

Consider a set of five data points where the $x$-values are $\{-2, -1, 0, 1, 2\}$ and the relationship is exactly $y = x^2$. What is the value of the Pearson's product-moment correlation coefficient $r$ for this dataset?

In a sports science study, the regression line of $y$ (athlete speed) on $x$ (leg strength) is $y = -0.8x + 15$. It is known that the variance of the leg strength scores is $25$ and the variance of the speed scores is $64$. Calculate the Pearson's product-moment correlation coefficient $r$.

If $r = 0.9$, the strength of the linear relationship is:

What is another name for the Pearson correlation coefficient?

If $\bar{x} = 5, \bar{y} = 15$ and the regression line equation is $y = ax + 5$, what is the value of $a$?

For the points $(0, 10), (1, 8), (2, 6)$, the correlation coefficient $r$ is:

Five different brands of laptops are ranked for their battery life and their processing speed by a tech reviewer. The results are shown in the following table:

• Laptop P: Battery Rank 1, Speed Rank 3
• Laptop Q: Battery Rank 2, Speed Rank 1
• Laptop R: Battery Rank 3, Speed Rank 2
• Laptop S: Battery Rank 4, Speed Rank 5
• Laptop T: Battery Rank 5, Speed Rank 4

Calculate the Spearman’s rank correlation coefficient, $r_s$, for this data using the formula $r_s = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}$.

A scientist is studying the relationship between the concentration of a chemical ($x$, in mg/L) and the reaction time ($y$, in seconds). The regression line of $y$ on $x$ is found to be $y = -1.2x + 25.4$. For a specific trial where the concentration was $10$ mg/L, the residual was calculated to be $-1.6$. Calculate the actual reaction time recorded for this trial.

The regression line of $y$ on $x$ is $y = ax + b$. If we know $r = 0.6$, $s_y = 10$, and $s_x = 5$, what is the value of the slope $a$?

A company models its monthly profit $P$ (in thousands of dollars) based on the number of employees $E$ using the line $P = 4.2E - 15$. If the current profit is 69 thousand dollars, how many employees does the model estimate the company has?

The regression line of $y$ on $x$ is $y = ax + b$. If we decrease every $x_i$ value by 5 and increase every $y_i$ value by 10, which of the following parameters of the regression line remains unchanged?

In a bivariate dataset, the covariance $S_{xy}$ is 48, the variance of $x$ is 64, and the variance of $y$ is 100. Calculate the Pearson correlation coefficient $r$.

If $A$ is an impossible event, then $P(A) =$

If the probability of winning a game is $\frac{2}{7}$, what is the probability of losing the game?

Given $U = \{1, 2, 3, 4, 5, 6\}$ and $A = \{2, 4, 6\}$, what is $P(A)$ if an element is chosen at random from $U$?

A fair coin is tossed. What is the probability of it landing on heads?

If $P(A) = 0.3$, $P(B) = 0.4$, and $P(A|B) = 0.5$, find $P(B|A)$.

A student takes two tests. The probability of passing the first is $0.8$ and the probability of passing the second is $0.7$. If these events are independent, find the probability of passing exactly one test.

In a Venn diagram, $P(A) = 0.5$, $P(B) = 0.4$, and $P(A' \cap B') = 0.3$. Find $P(A \cap B)$.

In a certain city, the probability of rainfall on any given day is $0.25$. A local events company offers a 'Rainy Day' insurance policy for a 5-day outdoor festival. The policy pays a fixed claim amount of $K$ USD to the organizers if it rains on 3 or more of the 5 days. To make the insurance policy sustainable, the company determines that the expected payout per policy should be exactly 20 USD. Find the required value of $K$, rounded to two decimal places.

A medical study categorizes participants into three age groups: Young ($40\%$), Middle-aged ($35\%$), and Senior ($25\%$). The probability of having a specific health condition is $0.05$ for the Young group and $0.12$ for the Middle-aged group. The probability for the Senior group is $p$. The overall probability that a randomly selected participant has the condition is $0.117$. Given that a participant has the condition, find the probability that they are in the Senior age group.

If $P(A) = 0.5, P(B) = 0.5$ and $P(A \cup B) = 0.75$, then $A$ and $B$ are:

In a bag of $12$ marbles, $4$ are red. If you draw two without replacement, what is the probability the second is red given the first was red?

If $P(A) = 0.2$, $P(B) = 0.3$, and $P(A \cap B) = 0.06$, are $A$ and $B$ independent?

Given $P(A) = 0.9$ and $P(B|A) = 0.9$, find $P(A \cap B)$.

In a class of $30$, $15$ play soccer and $10$ play tennis. $5$ play both. What is the probability a student plays soccer given they play tennis?

A Venn diagram shows $P(A \cap B) = 0.2$ and $P(A \cup B) = 0.8$. If $P(A) = P(B)$, find $P(A)$.

If $P(A) = 0.4$ and $P(B) = 0.3$, find $P(A \cup B)$ assuming $A$ and $B$ are independent.

There are two drawers. Drawer 1 has $2$ gold coins and $1$ silver coin. Drawer 2 has $1$ gold and $3$ silver coins. A drawer is chosen at random and a coin is picked. Find the probability the coin is gold.

A digital channel transmits bits 0 and 1. Due to static, $P(0 \text{ received} | 1 \text{ sent}) = p$ and $P(1 \text{ received} | 0 \text{ sent}) = p$. If $P(1 \text{ sent}) = 0.5$, for what value of $p$ is the received bit independent of the sent bit?

Let $A$ and $B$ be two events such that $P(A) = 0.6, P(B) = 0.3,$ and $P(A \cap B) = 0.2$. Find $P(A' | B')$.

A student takes a 3-question true/false test. They know the answer to the first question but guess on the other two. Given that they got at least two answers correct, what is the probability they got all three correct?

A bag contains $3$ red balls and $7$ blue balls. A ball is chosen at random, its color is recorded, and it is then replaced. This process is repeated $5$ times. Let $X$ be the number of red balls chosen. Find the expected value $E[X]$.

A spinner has three equal sections labeled 1, 2, and 3. It is spun $4$ times. Let $X$ be the number of times the spinner lands on 1. Find $P(X \le 1)$, rounding your answer to three decimal places.

The number of cars passing a specific point on a road follows a Poisson distribution with a mean of $2$ cars per minute. What is the expected number of cars to pass that point in a $5$-minute period?

A discrete random variable $X$ has a probability distribution given by $P(X=x) = kx$ for $x \in \{1, 2, 3\}$. Find the value of the constant $k$.

A call center receives two types of calls: sales and support. Sales calls arrive at a rate of $2.1$ per hour, and support calls arrive at a rate of $1.4$ per hour. Both follow independent Poisson processes. Find the probability that the center receives exactly $3$ calls in total during a randomly selected hour.

A game costs $5 to play. A fair six-sided die is rolled. If a $6$ is rolled, the player wins $20. If a $1$ or $2$ is rolled, the player wins $2. For any other outcome, the player wins nothing. Calculate the expected profit or loss for the player per game.

An online exam has a pass rate of $70\%$. If a group of $15$ students takes the exam independently, find the probability that at most $8$ of them pass.

A binomial distribution $X \sim B(n, p)$ has a mean of 6. Given that $P(X=0) = (0.7)^{20}$, calculate the value of $P(X=1)$.

A support desk has two independent phone lines. Line A receives an average of 3 calls per hour and Line B receives an average of 5 calls per hour. Find the probability that the support desk receives exactly 4 total calls in a 30-minute interval.

An exam consists of 20 multiple-choice questions. 14 of the questions are 'easy' and a student has a $90\%$ probability of answering each one correctly. The remaining 6 questions are 'hard' and the student has a $40\%$ probability of answering each one correctly. Find the expected total number of correct answers for this student.

If $P(Z > k) = 0.5$ in a standard normal distribution, what is $k$?

Given $X \sim N(40, 36)$, what is the value of $\mu$?

The Normal distribution is an example of:

If $X \sim B(6, 0.5)$, what is the most likely value of $X$ (the mode)?

If $X \sim B(n, p)$ and $n=5, p=0.1$, find $P(X=5)$.

A machine fills juice bottles. The volume is $N(500, 4)$. What percentage of bottles contains more than $502$ ml?

For a binomial distribution $X \sim B(n, p)$, if $n=10$ and $P(X=10) = (0.2)^{10}$, find $p$.

A company receives large batches of items. To check the quality, they take a random sample of 20 items from each batch. A batch is accepted if the sample contains at most 1 defective item. If the probability of a batch being accepted is 0.90, find the actual proportion of defective items, $p$, in the population to three significant figures.

In a game, a player tosses 10 fair coins simultaneously. This process is repeated 100 times. A 'jackpot' is defined as getting exactly 5 heads in a single toss of the 10 coins. Find the expected number of jackpots won across the 100 trials.

A manufacturer produces high-precision components with a 1% defect rate. These components are shipped in batches of size $n$. Quality control standards require that the probability of a batch containing two or more defective components must be no more than 0.05. Find the largest possible integer value of $n$.

A $z$-score of $1.5$ means the data value is:

If $X \sim N(30, 4)$, what is the value of $x$ that is two standard deviations above the mean?

What is the standard deviation of a standard normal distribution?

Which of the following is equivalent to $P(a < X < b)$ for a continuous random variable $X$?

Let $X \sim N(10, 4)$. Calculate $P(X > 7)$. (Use $P(Z < 1.5) = 0.9332$)

What is the $z$-score corresponding to the $1^{st}$ percentile of a normal distribution? (Given $P(Z < 2.326) = 0.99$)

In a distribution $N(50, 2^2)$, what is the probability that $X$ is exactly $50$?

A certain species of fish has lengths $L \sim N(25, \sigma^2)$ cm. If $20\%$ of the fish are longer than $30$ cm, find the probability that a fish is shorter than $15$ cm.