Question

Lefties A total of $11\%$of students at a large high school are left-handed. A statistics teacher selects a random sample of 100 students and records L= the number of left-handed students in the sample.

a. Explain why L can be modeled by a binomial distribution even though the sample was selected without replacement.

b. Use a binomial distribution to estimate the probability that 15 or more students in the sample are left-handed.

Short answer

(a)A binomial distribution can be used to model L.$\mathrm{L}~Bin(100,0.11)$

(b) The probability of greater than or equal to $15$ left-handed students in the sample is $0.13305$.

Step-by-step solution

Part (a) Step 1: Given Information

The number of left-handed pupils in the sample is denoted by the letter L.

The sample size is $100$.

In a large school, the percentage of left-handed kids is equal to $11$.

Part (a) Step 2: Simplification

The following concept was used:

$10\%$condition

$\mathrm{n}<0.10\mathrm{N}$

According to the rule, if the sample represents less than 10% of the population, it is safe to assume that the trials are independent and may be modelled using the binomial distribution, regardless of the without replacement sample.

The sample size$\left(=100\right)$is less than ten percent of a large school's students.

Furthermore, the left-handedness of one student has no bearing on the left-handedness of another, therefore each student's trial is independent.

Since, $L~Bin(100,0.11)$

As a result, a binomial distribution can be used to model L.

Part (b) Step 1: Given information

The number of left-handed pupils in the sample is denoted by the letter L.

The sample size is $100$.

In a large school, the percentage of left-handed kids is equal to $11$.

Part (b) Step 2: Simplification

Consider,

$$\begin{gathered} P(L\ge 15)=1-P(L<15) \\ P(L\ge 15)=1-P(L\le 14) \end{gathered}$$

Using TI- 83 plus

a) Now Press on $2^{nd}$and then click on Dist.

b) Go to binomcdf with $\downarrow$key.

c) Hit Enter

d) Type 100,0.11,14)

e) Hit Enter

The probability comes to be $0.86695$.

$$\begin{gathered} P(L\ge 15)=1-0.86695 \\ P(L\ge 15)=0.13305 \end{gathered}$$