Math Problem Statement
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]
Solution
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:
\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 weatherAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Contingency Tables
Independence of Events
Basic Probability
Formulas
Row total = sum of row values
Column total = sum of column values
Theorems
Independence of Events Theorem
Suitable Grade Level
Grades 6-8
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