Recent questions tagged statistics

A computer program receives a list of the $n$ distinct numbers $\{1,2, \ldots, n\}$ in a random order (i.e., all $n!$ permutations are equally likely). The program iterat...

The time $X$ (in minutes) a user spends on a website follows an exponential distribution with a mean of 0.5 minutes. The revenue $R$ generated from the user is a piecewis...

A loaded six-sided die has the following probability distribution for its outcome $X$ :$P(X=1)=0.1$ $P(X=2)=0.1$ $P(X=3)=0.1$ $P(X=4)=0.2$ $P(X=5)=0.2$ $P(X=6)=0.3$what i...

A university provost claims the average incoming student has an SAT score of $1200$. A researcher for the student newspaper is skeptical and believes the average score is...

A manufacturing process for engine pistons is specified to have a variance in piston diameter of $0.25 \mathrm{~mm}^2$. A new forging method is introduced, which is claim...

The standard deviation of a component in a drug is expected to be 0.00002 kg . A pharmacist suspecting the variability to be higher obtains a sample of 8 drugs and found ...

A new database query optimization algorithm is being tested. The old algorithm's mean execution time on a benchmark suite is 300 ms .A sample of 9 benchmark queries is ru...

A streaming service's data scientists know that $8 \%$ of users who visit a show's page will add it to their "Watchlist". After launching a new UI, they sample 2,000 user...

You keep tossing a fair coin until you see two consecutive heads (HH) for the first time. What is the expected number of tosses required?

Let $X_1, X_2, X_3$ be a random sample of size $3$ from a uniform distribution on the interval $[0,1]$.Let $Y=\max \left(X_1, X_2, X_3\right)$ be the maximum value in the...

A permutation $\pi=\left(\pi_1, \pi_2, \ldots, \pi_n\right)$ of the set $S=\{1,2, \ldots, n\}$ is chosen uniformly at random. A "fixed point" is defined as an index $i$ f...

Two distinct integers, $X$ and $Y$, are chosen at random without replacement from the set $S= \{1,2,3, \ldots, 100\}$.What is the variance of their sum, $\operatorname{Va...

In a class of 50 students, the number of students $n_i$ of each age $i$ is shown in the following table:\[\begin{array}{cr}\text{Age } (i) & n_i \\\hline18 & 20 \\19 & 22...

Two distinct integers, $X$ and $Y$, are chosen at random without replacement from the set $S= \{1,2,3, \ldots, 100\}$.What is the expected value of the smaller of the two...

The population standard deviation for the weight of 70-gram popcorn bags is claimed to be 2.5 grams. A random sample of 20 bags showed a sample standard deviation of 3 gr...

A car manufacturer advertises that its new model provides an average mileage of 40 km per litre. A consumer group is skeptical and believes the true mileage is less. To i...

Suppose that three random variables $X_1, X_2, X_3$ form a random sample from the uniform distribution on the interval $[0,1]$. Determine the value of$$E\left[\left(X_1-2...

Let $X$ and $Y$ have a continuous distribution with joint p.d.f.$$f(x, y)= \begin{cases}x+y & \text { for } 0 \leq x \leq 1 \text { and } 0 \leq y \leq 1 \\ 0 & \text { o...

Suppose that $X$ and $Y$ are random variables such that $\operatorname{Var}(X)=9, \operatorname{Var}(Y)=4$, and $\rho(X, Y)=-1 / 6$. Determine $\operatorname{Var}(X-3 Y+4...

Suppose that $X_1, \ldots, X_n$ form a random sample from a uniform distribution on the interval $[0, \theta]$, with a large sample size of $n=300$. The following hypothe...

A sociologist focusing on popular culture and media believes that the average number of hours per week (hrs/week) spent on social media is different for men and women. Th...

Suppose that people attending a party pour drinks from a bottle containing $63$ ounces of a certain liquid. Suppose also that the expected size of each drink is $2$ ounce...

Let $X_1, X_2, \ldots, X_{60}$ be a sequence of independent and identically distributed (i.i.d.) random variables, where $X_i$ represents the length of rope produced in t...

The probability that each specific child in a given family will inherit a certain disease is $0.2$. If it is known that at least one child in a family of $4$ children has...

Three men $A, B$, and $C$ shoot at a target. Suppose that $A$ shoots three times and the probability that he will hit the target on any given shot is $1 / 8, B$ shoots fi...

The IQs (intelligence quotients) of $25$ students from one batch of IITM students showed a mean of $110$ with a standard deviation of $8$, while the IQs of $25$ students ...

Suppose that $X, Y$, and $Z$ are three random variables such that $\operatorname{Var}(X)=1, \operatorname{Var}(Y)=4$, $\operatorname{Var}(Z)=8, \operatorname{Cov}(X, Y)=1...

Suppose that $X_1, \ldots, X_{10}$ are random variables such that the variance of each variable is $1$ and the correlation between each pair of different variables is $1 ...

It is claimed that the lifetimes of light bulbs are normally distributed with a mean of 800 hours and a standard deviation of 40 hours. We wish to test the hypothesis tha...

It is claimed that the lifetimes of light bulbs are normally distributed with a mean of $800$ hours and a standard deviation of $40$ hours. We wish to test the hypothesis...