A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think

Question
Two-way tables

A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think are the chances you will have much more than a middle-class income at age 30?" The two-way table summarizes the responses.
$$\begin{array}{c|cc|c} &\text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but } \ \text { probably not } & 426 & 286 & 712 \\\hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}$$
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Find $$P(C∣M)$$. Interpret this value in context.

2020-12-22
Definitions
Definition conditional probability:
$$P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{\text{P(A and B)}}{P(A)}$$
SOLUTION
$$\begin{array}{c|cc|c} &\text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but } \ \text { probably not } & 426 & 286 & 712 \\\hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}$$
G=Good chance
M=Male
We note that the table contains information about 4826 young adults (given in the bottom right corner of the table).
Moreover, 2459 of the 4826 young adults are male, becatise 2450 is mentioned in the row "Total” and in the column ”Male” of the table.
The probability is the number of favorable outcomes divided by the number of possible outcomes:
$$P(M)=\frac{\text{# of favorable outcomes}}{\text{of possible outcomes}}=\frac{2459}{4826}$$
Next, we note that. 758 of the 4826 children are from England and prefer Telepathy, because 758 is mentioned in the row ”A good chance” and in the column "Male” of the given table.
$$P(G \text{and} M)=\frac{\text{# of favorable outcomes}}{\text{of possible outcomes}}=\frac{758}{4826}$$
Use the definition of conditional probability:
$$P(G|M)=\frac{P(G \text{and} M)}{P(M)} = \frac{\frac{758}{4826}}{\frac{2459}{4826}}=\frac{758}{2459}\approx0.3083=30.83\%$$
30.83% of the male young adults think there is a good chance that they will have much more than a middle-class income at age 30.
$$\frac{758}{2459}\approx0.3083=30.83\%$$

Relevant Questions

A survey of 4826 randomly selected young adults (aged 19 to 25 ) asked, "What do you think are the chances you will have much more than a middle-class income at age 30?" The two-way table summarizes the responses. $$\begin{array} {lc} & \text{Gender} \ \text {Opinion} & \begin{array}{l|c|c|c} & Female & Male & Total \\ \hline \text{Almost no chance} & 96 & 98 & 194 \\ \hline \begin{array}{l} \text{Some chance but} \\ \text{probably not} \end{array} & 426 & 286 & 712 \\ \hline A\ 50-50\ chance & 696 & 720 & 1416 \\ \hline \text{A good chance} & 663 & 758 & 1421 \\ \hline \text{Almost certain} & 486 & 597 & 1083 \\ \hline Total & 2367 & 2459 & 4826 \end{array}\ \end{array}$$

Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Find P(G | M). Interpret this value in context.

A random sample of U.S. adults was recently asked, "Would you support or oppose major new spending by the federal government that would help undergraduates pay tuition at public colleges without needing loans?" The two-way table shows the responses, grouped by age.

$$\begin{array}{ccc} & Age \ Response & {\begin{array}{l|r|r|r|r|r} & 18-34 & 35-49 & 50-64 & 65+ & Total \\ \hline Support & 91 & 161 & 272 & 332 & 856 \\ \hline Oppose & 25 & 74 & 211 & 255 & 565 \\ \hline Don't know & 4 & 13 & 20 & 51 & 88 \\ \hline Total & 120 & 248 & 503 & 638 & 1509 \end{array}} \ \end{array}$$

Do these data provide convincing evidence of an association between age and opinion about loan-free tuition in the population of U.S. adults?

A survey of 4826 randomly selected young adults (aged 19 to 25 ) asked, "What do you think are the chances you will have much more than a middle-class income at age 30? The two-way table summarizes the responses.
$$\begin{array} {c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but probably not } & 426 & 286 & 712 \\ \hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}$$
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Given that the chosen student didn't say "almost no chance," what's the probability that this person is female? Write your answer as a probability statement using correct symbols for the events.

A random sample of 88 U.S. 11th- and 12th-graders was selected. The two-way table summarizes the gender of the students and their response to the question "Do you have allergies?" Suppose we choose a student from this group at random.

$$\begin{array}{c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text{ Yes } & 19 & 15 & 34 \\ \text{ No } & 24 & 30 & 54 \\ \hline \text{ Total } & 43 & 45 & 88\\ \end{array}$$
What is the probability that the student is female or has allergies?
$$(a)\frac{19}{88}$$
(b)$$\frac{39}{88}$$
(c)$$\frac{58}{88}$$
(d)$$\frac{77}{88}$$

Using data from the 2000 census, a random sample of 348 U.S. residents aged 18 and older was selected. The two-way table summarizes the relationship between marital status and housing status for these resident.

$$\begin{array}{l|c|c|c} & Married & Not married & Total \\ \hline Own & 172 & 82 & 254 \\ \hline Rent & 40 & 54 & 94 \\ \hline Total & 212 & 136 & 348 \end{array}$$

State the hypotheses for a test of the relationship between marital status and housing status for U.S. residents.

Statistics students at a state college compiled the following two-way table from a sample of randomly selected students at their college:
$$\begin{array}{|c|c|c|}\hline&\text{Play chess}&\text{Don`t play chess}\\\hline \text{Male students} &25&162\\ \hline \text{Female students}&19&148 \\ \hline \end{array}\\$$
Answer the following questions about the table. Be sure to show any calculations.
What question about the population of students at the state college would this table attempt to answer?
State $$H^0$$ and $$H^1$$ for the test related to this table.

The following two-way contingency table gives the breakdown of the population of adults in a town according to their highest level of education and whether or not they regularly take vitamins:
$$\begin{array}{|c|c|c|c|c|} \hline \text {Education}& \text {Use of vitamins takes} &\text{Does not take}\\ \hline \text {No High School Diploma} & 0.03 & 0.07 \\ \hline \text{High School Diploma} & 0.11 & 0.39 \\ \hline \text {Undergraduate Degree} & 0.09 & 0.27 \\ \hline \text {Graduate Degree} & 0.02 & 0.02 \\ \hline \end{array}$$
You select a person at random. What is the probability the person does not take vitamins regularly?

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. This two-way table summarizes the survey data.

$$\begin{array} {|c|} & Grade \ Most important & \begin{array}{l|c|c|c|c} & \begin{array}{c} 4 \mathrm{th} \ grade \end{array} & \begin{array}{c} 5 \mathrm{th} \ \text { grade } \end{array} & \begin{array}{c} 6 \mathrm{th} \ grade \end{array} & Total \\ \hline Grades & 49 & 50 & 69 & 168 \\ \hline Athletic & 24 & 36 & 38 & 98 \\ \hline Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array} \ \end{array}$$

Suppose we select one of these students at random. What's the probability that: The student is a sixth grader or a student who rated good grades as important?

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.

$$Grade\ Most\ important\begin{array}{l|c|c|c|c} & 4 \mathrm{th} & 5 \mathrm{th} & 6 \mathrm{th} & \text { Total } \\ \hline Grades & 49 & 50 & 69 & 168 \\ \hline Athletic & 24 & 36 & 38 & 98 \\ \hline Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array}$$

Suppose we select one of these students at random. Find P(athletic | 5th grade).

The two-way table shows the results from a survey of dog and cat owners about whether their pet prefers dry food or wet food.
$$\begin{array}{c|c|c} &\text { Dry } & \text { Wet }\\ \hline \text{ Cats } & 10&30 \\ \hline \text{ Dogs} & 20&20\\ \end{array}$$
Does the two-way table show any difference in preferences between dogs and cats? Explain.

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