Question

The Director of Admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience. What is the expected number of admissions for the fall semester? Compute the variance and the standard deviation of the number of admissions.

Admissions Probability
1,000 .6
1,200 .3
1,500 .1

Answer 1

Concepts and reason

Mean:

Mean is the most commonly used measure of central tendency. It is the average of all the observations in the data set. In other words, the mean is obtained by summing the observed numerical values of a variable in a set of data and then dividing the total by the number of observations involved.

Expected value (mean) of probability distribution:

The average of the all the possible outcomes of the random variable is called as the expected value of the probability distribution. The outcomes for the random variable can be either the probability outcomes or the relative frequencies of the variable.

Standard deviation:

The mean is the average of all samples. The variance is the average of the squared differences from the mean. And the standard deviation refers to the square root of the variance. Standard deviation refers to how closely the samples are located to the mean. In other words, standard deviation refers to how much the values in the distribution vary from the mean value of that distribution. Larger the variability around the mean, the larger will be the standard deviation.

The variation between each observation from its mean is known as standard deviation. The greater the distance of the points from the mean, the greater is the variability and vice-versa.

Fundamentals

Discrete mean and standard deviation:

• The formula for finding $E\left( x \right)$ is, $E\left( x \right) = \sum {xP\left( x \right)}$ .

• Formula for finding $E\left( {{x^2}} \right)$ is $E\left( {{x^2}} \right) = \sum {{x^2}P\left( x \right)}$ .

• Formula for finding the variance of x is, $V\left( x \right) = E\left( {{x^2}} \right) - {\left\{ {E\left( x \right)} \right\}^2}$ .

• Formula for finding the standard deviation of x is, $SD\left( x \right) = \sqrt {E\left( {{x^2}} \right) - {{\left\{ {E\left( x \right)} \right\}}^2}}$ .

The expected number of admissions for the fall semester is obtained below:

From the given information, student admissions for the fall semester on the basis of past experience and their probabilities are obtained. Let x denotes the number of admissions for the fall semester.

The required probabilities are,

The expected value is,

$\begin{array}{c}\\E\left( x \right) = 1,000\left( {0.6} \right) + 1,200\left( {0.3} \right) + 1,500\left( {0.10} \right)\\\\ = 600 + 360 + 150\\\\ = 1,110\\\end{array}$

The value of $E\left( {{x^2}} \right)$ is obtained below:

The value for $E\left( {{x^2}} \right)$ is,

$\begin{array}{c}\\E\left( {{x^2}} \right) = 1,{000^2}\left( {0.6} \right) + 1,{200^2}\left( {0.3} \right) + 1,{500^2}\left( {0.10} \right)\\\\ = 600,000 + 432,000 + 225,000\\\\ = 1,257,000\\\end{array}$

The variance of the number of admissions is obtained below:

The required variance is,

$\begin{array}{c}\\V\left( x \right) = 1,257,700 - {\left( {1,110} \right)^2}\\\\ = 1,257,000 - 1,232,100\\\\ = 24,900\\\end{array}$

The standard deviation of the number of admissions is obtained below:

The required standard deviation is,

$\begin{array}{c}\\SD\left( x \right) = \sqrt {1,257,700 - {{\left( {1,110} \right)}^2}} \\\\ = \sqrt {1,257,000 - 1,232,100} \\\\ = \sqrt {24,900} \\\\ = 157.797\\\end{array}$

Ans:

The expected number of admissions for the fall semester is 1,110.

The variance of the number of admissions is 24,900.

The standard deviation of the number of admissions is 157.797.