The table below summarizes results from a study of people who refused to answer survey questions. Use the table to answer the following questions. Be very careful to read the question carefully to determine whether or not there is overlap.

Age | 18-21 | 22-29 | 30-39 | 40-49 | 50-59 | 60 and over |
---|---|---|---|---|---|---|

Responded | 70 | 252 | 242 | 133 | 135 | 199 |

Refused | 13 | 22 | 35 | 28 | 37 | 59 |

If one of the subjects is randomly selected, what is the probability that a randomly selected person is between 22 and 29 years or refused to answer?

The probability that a randomly selected person is between 22 and 29 years or refused to answer is _____

(Do not round until the final answer. Then round to three decimal places as needed.

If one of the subjects is randomly selected, what is the probability that the selected person was between 16 and 21 years or refused to answer?

The probability that the selected person was between 18 and 21 year or refused to answer is _____

(Do not round until the final answer. Then round to three decimal places as needed.)

### Solution

Age | 18-21 | 22-29 | 30-39 | 40-49 | 50-59 | 60 and over |
---|---|---|---|---|---|---|

Responded | 70 | 252 | 242 | 133 | 135 | 199 |

Refused | 13 | 22 | 35 | 28 | 37 | 59 |

Let $A$ denote person is between 22 and 29, $B$ denote denote the person refused to answer.

$P(A) = \frac{274}{1225}$, $P(B) = \frac{194}{1225}$ and $P(A\cap B) =\frac{22}{1225}$.

The probability that the selected person was between 22 and 29 year or refused to answer is $P(A\cup B)$

` $$ \begin{aligned} P(A\cup B) &= P(A) + P(B) - P(A\cap B)\\ &= \frac{274}{1225} + \frac{194}{1225} - \frac{22}{1225}\\ &= 0.364 \end{aligned} $$ `

The probability that a randomly selected person is between 22 and 29 years or refused to answer is **0.364**

Let $C$ denote person is between 18 and 21, $D$ denote denote the person refused to answer.

$P(C) = \frac{83}{1225}$, $P(D) = \frac{194}{1225}$ and $P(C\cap D) =\frac{13}{1225}$.

The probability that the selected person was between 18 and 21 year or refused to answer is $P(C\cup D)$

` $$ \begin{aligned} P(C\cup D) &= P(C) + P(D) - P(C\cap D)\\ &= \frac{83}{1225} + \frac{194}{1225} - \frac{13}{1225}\\ &= 0.216 \end{aligned} $$ `

The probability that a randomly selected person is between 18 and 21 years or refused to answer is **0.216**

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators