### The joint density of x and y is f(x y)=c(x^2-y^2)e^{-x}. Find the conditional distribution of Y, given X=x.

[ f(x, y) = c (x^2 – y^2) hspace{0.5in} 0 leq x lt infty, hspace{0.2in} -x leq y leq x ] This question aims to find the conditional distribution of the given function with a given condition X=x. The question is based on the joint density function and conditional distribution concepts. The conditional distribution is the […]

### Prove that if m and n are integers and m x n is even, then m is even or n is even.

This problem aims to familiarize us with the method of poof. The concept required to solve this problem is related to discrete mathematics, including direct proof or proof by contradiction, and proof by contrapositive. There are multiple methods to write a proof, but here we are going to see only two methods, proof by contradiction […]

### For a test of Ho: p=0.5,the z test statistic equals -1.74. Find the p-value for Ha: p<0.5.

The question aims to find out the p-value using the given alternative hypothesis, which is a one-sided hypothesis. Therefore, the p-value will be determined for the left tail test with reference to the standard normal probability table. When the alternative hypothesis states that a certain value for a parameter in the null hypothesis is lesser […]

### Suppose that X is a normal random variable with mean 5. If P(X>9)=0.2, approximately what is Var(X)?

This question aims to find the probability of a normally distributed random variable $X$. A random variable is one whose value is determined by the results of a statistical experiment. The normal distribution, also known as the Gaussian distribution or the z-distribution, has a mean of zero and a standard deviation of one. Data in a […]

### Suppose that the height in inches of a 25-year-old man is a normal random variable with parameters μ=71 and σ^2=6.25.

-a) What percentage of 25-year-old men are over $6$ feet, $2$ inches tall? -b) What percentage of men in the $6$-footer club are over $6$ feet, $5$ inches? This question aims to explain the mean, variance, standard deviation, and z-score. The mean is the central or the most common value in a group of numbers. […]

### Find the critical value z a/2 that corresponds to a 93% confidence level.

This question belongs to the statistics domain and aims to understand the alpha level, confidence level, z-critical values, the term $z_{alpha /2}$ and further explains how to calculate these parameters. The alpha level or significance level is the probability of producing a false decision when the null hypothesis is correct. Alpha levels are employed in […]

### What is an advantage of using a stem-and-leaf plot instead of a histogram? What is the disadvantage?

This question aims to resolve the advantages and disadvantages of using a stem-and-leaf plot for visualizing statistical data. Stem-and-leaf plots are used widely in visualizing the overall summary of statistical data. To develop an understanding of the core concept, let’s consider the following arbitrary data as an example: { 1, 2, 3, 4, 5, 11, […]

### How many ways are there to distribute six indistinguishable balls into nine distinguishable bins?

The objective of this question is to find the number of ways the six indistinguishable balls can be distributed into nine distinguishable bins. A mathematical method for determining the number of potential groupings in a set of objects in which the selection order becomes irrelevant is referred to as combination. The objects can be chosen […]

### A mail-order company advertises that it ships 90% of its orders within three working days. You select an SRS of 100 of the 5000 orders received in the past week for an audit. The audit reveals that 86 of these orders were shipped on time. If the company really ships 90% of its orders on time, what is the probability that the proportion in an SRS of 100 orders is 0.86 or less?

This question widely explains the concept of the sampling distribution of sample proportions. The population proportion plays an important role in many areas of science. This is because research questionnaires in many fields involve this parameter. The success proportion is calculated by the sampling distribution of sample proportions. It is the ratio of the chance […]

### Let x represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?

The aim of this question is to understand the key concept of a random variable using the coin toss experiment which is the most basic binomial (experiment with two possible outcomes) experiment performed in probability theory. A random variable is nothing but a mathematical formula used to describe the outcome of statistical experiments. For example, […]