A survey of 1 comma 0911,091 adults was​ asked, "Do you feel overloaded with too much​ information?" Suppose that the results indicated that of 542542 ​males, 238238 answered yes. Of 549549 ​females, 275275 answered yes. Construct a contingency table to evaluate the probabilities. Complete parts​ (a) through​ (d).
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Part 1
a. What is the probability that a respondent chosen at random indicates that​ he/she feels overloaded with too much​ information?
0.4700.470 ​(Round to three decimal places as​ needed.)
Part 2
b. What is the probability that a respondent chosen at random is a female and indicates that​ he/she feels overloaded with too much​ information?
enter your response here ​(Round to three decimal places as​ needed.)

To solve this, we need to understand the question and use a contingency table format to calculate probabilities. The total number of participants in the survey is 1,091 adults, with 542 males and 549 females. We also know how many answered "yes" to the question of feeling overloaded with too much information.

Let's break it down:

### Step 1: Contingency Table
We create a table that separates the responses ("Yes" and "No") by gender.

| Gender  | Yes  | No   | Total |
|---------|------|------|-------|
| Male    | 238  | 542 - 238 = 304  | 542   |
| Female  | 275  | 549 - 275 = 274  | 549   |
| **Total**| 238 + 275 = 513 | 304 + 274 = 578 | 1,091  |

Now let's tackle each part.

### Part (a)
**What is the probability that a respondent chosen at random indicates that they feel overloaded with too much information?**

This is asking for the probability of someone responding "yes," regardless of gender. We can calculate this by dividing the total number of people who responded "yes" by the total number of respondents.

\[
P(\text{Yes}) = \frac{513}{1,091} = 0.470 \text{ (rounded to three decimal places)}
\]

This matches the given answer for part (a).

### Part (b)
**What is the probability that a respondent chosen at random is a female and indicates that they feel overloaded with too much information?**

Here, we need to find the probability that the selected respondent is both female and answered "yes." This is calculated by dividing the number of females who answered "yes" by the total number of respondents.

\[
P(\text{Female and Yes}) = \frac{275}{1,091}
\]

\[
P(\text{Female and Yes}) = 0.252 \text{ (rounded to three decimal places)}
\]

Thus, the probability that a randomly chosen respondent is a female and feels overloaded with too much information is 0.252.

Would you like to proceed with parts (c) and (d), or do you have any questions so far?

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**Here are five additional questions related to this topic:**
1. What is the probability that a randomly chosen male answered "yes"?
2. What is the probability that a randomly chosen respondent is a male and answered "no"?
3. What is the probability that a respondent is female, given that they answered "yes"?
4. What is the probability that a respondent feels overloaded, given that they are female?
5. What is the probability that a respondent did not feel overloaded with information?

**Tip:** In problems like this, ensure you understand how the total sample space changes based on conditional probabilities!

A survey of 1,091 adults was asked, 'Do you feel overloaded with too much information?' Suppose that the results indicated that of 542 males, 238 answered yes. Of 549 females, 275 answered yes. Construct a contingency table to evaluate the probabilities. Complete parts (a) through (d).

Analyze survey data from 1,091 adults, where 238 males and 275 females felt overloaded with too much information. Learn how to construct a contingency table and calculate the probabilities of different outcomes, including conditional probabilities. Suitable for Grades 9-12 students studying probability.

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