Question

Do you think global warming will have an impact on you during your lifetime? A CBS News/New York Times poll of 1000 adults in the United States asked this question (CBS News website, December, 2014). Consider the responses by age groups shown below.

Response Yes Unsure Age 30+ 18-29 293 134 131 432

(Incase you can't see the above image)

Response --- 18-29 --- 30+

Yes ------------ 134 ------293

No ------------- 131 ------432

Unsure -------- 2 -----------8

a. What is the probability that a respondent 18-29 years of age thinks that global warming will not pose a serious threat during his/her lifetime (to 4 decimals)?

b. What is the probability that a respondent 30+ years of age thinks that global warming will not pose a serious threat during his/her lifetime (to 4 decimals)?

c. For a randomly selected respondent, what is the probability that a respondent answers yes (to 3 decimals)?

d. Based on the survey results, does there appear to be a difference between ages 18-29 and 30+ regarding concern over global warming?

Answer 1

Concepts and reason

Probability:

The ratio of the number of favorable outcomes to certain event and the total number of possible outcomes is called as the probability of an event.

Complementary events:

Complementary of an event represents not happening of that event. That is, if an event A is defined, then complement of event A is not occurring of event A.

Fundamentals

The probability of an event is defined as,

$\begin{array}{c}\\{\rm{Probability}} = \frac{{{\rm{Number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}\,{\rm{for}}\,{\rm{an}}\,{\rm{event}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes}}}}\\\\ = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\end{array}$

Compliment rule:

$P\left( {{A^C}} \right) = 1 - P\left( A \right)$

Formula for finding the value of $P\left( {X \le x} \right)$ is, $P\left( {X \le x} \right) = 1 - P\left( {X > X} \right)$

(a)

The probability that a respondent 18-29 years of age thinks that global warming will not pose a serious threat during his/her lifetime is obtained below:

From the given information, in an Age group 18-29 134 responded that yes, 131 responded that No and 2 were responded that unsure. In age group $30 +$ 293 were responded that yes, 432 were responded that No and 8 were responded that Unsure.

The required probability is,

$\begin{array}{c}\\P\left( \begin{array}{l}\\{\rm{Respondent 18 - 29 years}}\\\\{\rm{ of age thinks that global}}\\\\{\rm{ warming will not pose a}}\\\\{\rm{ serious threat during his/her}}\\\\{\rm{ lifetime}}\\\end{array} \right) = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\\ = \left( {\frac{\begin{array}{l}\\{\rm{Number}}\,{\rm{of}}\,{\rm{respondedts}}\\\\\,{\rm{of}}\,{\rm{age}}\,{\rm{18 - 29}}\,{\rm{who}}\,{\rm{said}}\,{\rm{NO}}\\\end{array}}{\begin{array}{l}\\{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\\\\\,{\rm{18 - 29}}\,{\rm{age}}\,\\\\{\rm{respondents}}\\\end{array}}} \right)\\\\ = \left( {\frac{{131}}{{131 + 134 + 2}}} \right)\\\\ = \frac{{131}}{{267}}\\\end{array}$

$= 0.4906$

(b)

The probability that a respondent 30+ years of age thinks that global warming will not pose a serious threat during his/her lifetime is obtained below:

The required probability is,

$\begin{array}{c}\\P\left( \begin{array}{l}\\{\rm{Respondent 30 + years}}\\\\{\rm{ of age thinks that global}}\\\\{\rm{ warming will not pose}}\\\\{\rm{ a serious threat}}\\\\{\rm{ during his/her lifetime}}\\\end{array} \right) = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\\ = \left( {\frac{\begin{array}{l}\\{\rm{Number}}\,{\rm{of}}\,{\rm{respondedts}}\\\\\,{\rm{of}}\,{\rm{age}}\,30 + {\rm{who}}\,{\rm{said}}\,{\rm{NO}}\\\end{array}}{\begin{array}{l}\\{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\\\\\,30 + \,{\rm{age}}\,\\\\{\rm{respondents}}\\\end{array}}} \right)\\\\ = \left( {\frac{{432}}{{293 + 432 + 8}}} \right)\\\\ = \frac{{432}}{{733}}\\\end{array}$

$= 0.5894$

(c)

The probability that a respondent answers yes is obtained below:

The required probability is,

$\begin{array}{c}\\P\left( {{\rm{A Respondent answers yes}}} \right) = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\\ = \left( {\frac{{{\rm{Total Number}}\,{\rm{of}}\,{\rm{respondedts who said yes}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of respondents}}}}} \right)\\\\ = \left( {\frac{{134 + 293}}{{134 + 131 + 2 + 293 + 432 + 8}}} \right)\\\\ = \frac{{427}}{{1,000}}\\\end{array}$

$= 0.427$

(d)

The difference between ages 18-29 and 30+ regarding concern over global warming is obtained below:

The probability that a respondent 18-29 years of age thinks that global warming will not pose a serious threat during his/her lifetime is,

$\begin{array}{c}\\P\left( \begin{array}{l}\\{\rm{Respondent 18 - 29 years}}\\\\{\rm{ of age thinks that global}}\\\\{\rm{ warming will not pose a}}\\\\{\rm{ serious threat during his/her}}\\\\{\rm{ lifetime}}\\\end{array} \right) = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\\ = \left( {\frac{\begin{array}{l}\\{\rm{Number}}\,{\rm{of}}\,{\rm{respondedts}}\\\\\,{\rm{of}}\,{\rm{age}}\,{\rm{18 - 29}}\,{\rm{who}}\,{\rm{said}}\,{\rm{NO}}\\\end{array}}{\begin{array}{l}\\{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\\\\\,{\rm{18 - 29}}\,{\rm{age}}\,\\\\{\rm{respondents}}\\\end{array}}} \right)\\\\ = \left( {\frac{{131}}{{131 + 134 + 2}}} \right)\\\\ = \frac{{131}}{{267}}\\\end{array}$

$= 0.4906$

The probability that a respondent 30+ years of age thinks that global warming will not pose a serious threat during his/her lifetime is,

$\begin{array}{c}\\P\left( \begin{array}{l}\\{\rm{Respondent 30 + years}}\\\\{\rm{ of age thinks that global}}\\\\{\rm{ warming will not pose}}\\\\{\rm{ a serious threat}}\\\\{\rm{ during his/her lifetime}}\\\end{array} \right) = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\\ = \left( {\frac{\begin{array}{l}\\{\rm{Number}}\,{\rm{of}}\,{\rm{respondedts}}\\\\\,{\rm{of}}\,{\rm{age}}\,30 + {\rm{who}}\,{\rm{said}}\,{\rm{NO}}\\\end{array}}{\begin{array}{l}\\{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\\\\\,30 + \,{\rm{age}}\,\\\\{\rm{respondents}}\\\end{array}}} \right)\\\\ = \left( {\frac{{432}}{{293 + 432 + 8}}} \right)\\\\ = \frac{{432}}{{733}}\\\end{array}$

$= 0.5894$

Difference is,

$\begin{array}{c}\\{\rm{Difference}} = P\left( \begin{array}{l}\\{\rm{Respondent 18 - 29 years}}\\\\{\rm{ of age thinks that global}}\\\\{\rm{ warming will not pose a}}\\\\{\rm{ serious threat during his/her}}\\\\{\rm{ lifetime}}\\\end{array} \right) \sim P\left( \begin{array}{l}\\{\rm{Respondent 30 + years}}\\\\{\rm{ of age thinks that global}}\\\\{\rm{ warming will not pose}}\\\\{\rm{ a serious threat}}\\\\{\rm{ during his/her lifetime}}\\\end{array} \right)\\\\ = 0.4906 \sim 0.5894\\\\ = 0.099\\\end{array}$

There is a difference between ages 18-29 and 30+ regarding concern over global warming.

Ans: Part a

The probability that a respondent 18-29 years of age thinks that global warming will not pose a serious threat during his/her lifetime is 0.4906.

Part b

The probability that a respondent 30+ years of age thinks that global warming will not pose a serious threat during his/her lifetime is 0.5894.

Part c

The probability that a respondent answers yes 0.427.

Part d

Yes. There is a difference between ages 18-29 and 30+ regarding concern over global warming.