One rainy Saturday morning, Adam woke up to hear his mom complaining about the house being dirty. "Mom is always grouchy when it rains," Adam's brother said to him.
Adam decided to figure out if this statement was actually true. For the next month, he charted every time it rained and every time his mom was grouchy. He found that rainy days and his mom being grouchy were entirely independent events. Some of his data are shown in the table below.
Fill in the missing values from the frequency table.
Raining	Not raining	Row total
Grouchy	
\[3\]
Not grouchy	
\[27\]
Column total	
\[10\]	
\[20\]	
\[30\]


We are given a contingency table with some missing values. The table represents the frequency of events based on two variables: whether it is raining or not raining, and whether Adam's mom is grouchy or not grouchy. Let's fill in the missing values step by step.

Here is the table with some values:

\[
\begin{array}{|c|c|c|c|}
\hline
\text{Event} & \text{Raining} & \text{Not raining} & \text{Row total} \\
\hline
\text{Grouchy} & 3 & ? & ? \\
\text{Not grouchy} & ? & 27 & ? \\
\hline
\text{Column total} & 10 & 20 & 30 \\
\hline
\end{array}
\]

### Step 1: Fill in the missing row totals
We know that the column totals are given as 10 for "Raining" and 20 for "Not raining," with a grand total of 30. The sum of the row totals must also equal 30.

Let \( a \) be the total number of times Adam's mom was grouchy, and \( b \) be the total number of times she was not grouchy.

So:
\[
a + b = 30
\]

From the column total, we know:
\[
b = 30 - a
\]

### Step 2: Use the known grouchy data
We are given that on 3 rainy days, Adam's mom was grouchy. Therefore, the remaining grouchy days when it was not raining is:
\[
a = 3 + x
\]

Continuuitable use outside chair weather

One rainy Saturday morning, Adam woke up to hear his mom complaining about the house being dirty. 'Mom is always grouchy when it rains,' Adam's brother said to him. Adam decided to figure out if this statement was actually true. For the next month, he charted every time it rained and every time his mom was grouchy. He found that rainy days and his mom being grouchy were entirely independent events. Some of his data are shown in the table below. Fill in the missing values from the frequency table. Raining Not raining Row total Grouchy [3] Not grouchy [27] Column total [10] [20] [30]

This problem involves solving a contingency table by filling in missing values based on given data about the independence of events. Explore how to complete the table using row and column totals, basic probability concepts, and the independence of events theorem. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation