Question

Page of 12 Binomial Experiments Previously, we learned about binomial experiments. A binomial experiment consists of n independent trials, each having two possible outcomes: success, and failure. In addition, we define p to be the probability of success in one trial, and x is the number of successes in n trials. The probability of obtaining x successes is denoted P(x). The formula for computing this is P(x) = C:p. (1 - p)"-* In this lesson, we use technology rather than the formula to compute the binomial probabilities. There are various online applets, statistics software packages and calculators that can be utilized. One such tool is an online applet available at the address below: https://carnegiemathpathways.org/go/Binomial TRY THESE Suppose that the president has a 45% approval rate, and that you survey 10 people randomly to ask them whether they approve of the president. Let x represent the number of registered voters who approve of the president. Explain why x has a binomial distribution, then identify n and p. 1 Statway Version 3.1. © 2018 WestEd. All rights reserved. 6.2.41

Answer 1

Q1) We are given here that:
P(approval rate) = 0.45

and sample size n = 10 here.

The number of registered voters who approve of the president is modelled here as:

$X \sim Bin(n = 10, p= 0.45)$

This is a binomial distribution, because the probability of approval is same for all voters and independent of each other. Also the number of trials here is constant = 10.

Q2) The probabilities here are computed as:

$P(X =x) = \binom{10}{x}0.45^x(1 - 0.45)^{10- x}$

Using this the probabilities here are computed as:

x	p(x)
0	0.0025
1	0.0207
2	0.0763
3	0.1665
4	0.2384
5	0.2340
6	0.1596
7	0.0746
8	0.0229
9	0.0042
10	0.0003

Q3) The mean here is computed as:
Mean = np = 10*0.45 = 4.5

Therefore 4.5 is the required mean value here.

Q4) The proportion and percentage here is computed as:

proportion = 6/10 = 0.6

Percentage = 60%