Question

A sample of 20 students who had recently taken elementary statistics yielded the following informatin on brand of calculator owned (T = Texas instruments, H = Hewlett Packard, C = Casio, S = Sharp):

C S T C C S T T S C

C T T T H H S S C H

a. Estimate the true proportion of all such students who own a Texas Instruments calculator.

b. Of the 6 students who owned a TI calculator, 2 had graphing calculators. Estimate the proportion of students who do not own a TI graphing calculator.

Answer 1

Concepts and reason

The concept used in this problem is the concept of the proportions.

The proportion can be defined as the percentage of the success or failure which associated with the study population or the sample.

Fundamentals

Consider that a certain set contain total $N$ elements and out of those elements $n$ belong to a certain category considering $n < N$ . Then the proportion elements that belongs to the particular category can be obtained as:

$p = \frac{n}{N}$

The proportion of all those elements which do not belong to that certain category can be obtained as:

$\begin{array}{c}\\q = 1 - p\\\\ = 1 - \frac{n}{N}\\\end{array}$

(a)

The sample of twenty students are provided. They use different calculators and it can be written as follows:

$\left\{ \begin{array}{l}\\C,S,T,C,C,S,T,T,S,C\\\\C,T,T,T,H,H,S,S,C,H\\\end{array} \right\}$

In the provided set, C, H, T, and S represents the different type of calculators as CASLO, Hewlett, Texas, and Sharp respectively. The number of all of those calculators has been provided below:

$\begin{array}{l}\\C = 6\\\\S = 5\\\\H = 3\\\\T = 6\\\end{array}$

The proportion of all those students who uses the Texas calculator can be found as:

$\begin{array}{c}\\p = \frac{n}{N}\\\\ = \frac{6}{{20}}\\\\ = 0.3\\\end{array}$

It is required to obtain the proportion of people who does not use the TI calculator.

The proportion of all those students who own the Texas calculator can be found as:

$\begin{array}{c}\\p = \frac{n}{N}\\\\ = \frac{2}{{20}}\\\\ = 0.1\\\end{array}$

The estimate of the proportion of students who do not own a TI graphing calculator is,

$\begin{array}{c}\\p = 1 - \frac{n}{N}\\\\ = 1 - 0.1\\\\ = 0.9\\\end{array}$

The above calculation is quite intuitive. The proportion of some category and proportion of complement of that category will add up to one.

Ans:

The proportion of students who do not own a TI graphing calculator is $0.9$ .