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Who eats breakfast? Students in an urban school were curious about how many children regularly eat breakfast. They conducted a survey, asking, "Do you eat breakfast on a regular basis?" All 595 students in the school responded to the survey. The resulting data are shown in the two-way table below. $$\begin{array}{lccc}\hline & \text { Male } & \text { Female } & \text { Total } \\ \text { Eats breakast regularly } & 190 & 110 & 300 \\\\\text { Doesn't eat breaktast regularly } & 130 & 165 & 295 \\\\\text { Total } & 320 & 275 & 595 \\\\\hline\end{array}$$ If we select a student from the school at random, what is the probability that the student is (a) a female? (b) someone who eats breakfast regularly? (c) a female and eats breakfast regularly? (d) a female or eats breakfast regularly?

Step by step solution

01

Identify Total Number of Students

From the two-way table, observe that there is a total of 595 students in the school.

02

Calculate Probability of (a) a Female

The total number of female students is 275. The probability that a student is a female is calculated by dividing the number of females by the total number of students: \( \frac{275}{595} \). Simplifying this fraction: \( \frac{11}{24} \).

03

Calculate Probability of (b) Eats Breakfast Regularly

The total number of students who eat breakfast regularly is 300. The probability that a student eats breakfast regularly is \( \frac{300}{595} \). Simplification gives: \( \frac{60}{119} \).

04

Calculate Probability of (c) a Female and Eats Breakfast Regularly

From the table, 110 female students eat breakfast regularly. Therefore, the probability is \( \frac{110}{595} \). Simplifying the fraction gives \( \frac{22}{119} \).

05

Calculate Probability of (d) a Female or Eats Breakfast Regularly

To find this probability, add the probability of a female and the probability of eating breakfast regularly, then subtract the probability of both events happening simultaneously: \[P(F \cup H) = P(F) + P(H) - P(F \cap H)\]Substitute the probabilities: \[\frac{275}{595} + \frac{300}{595} - \frac{110}{595} = \frac{465}{595} = \frac{93}{119}\] as simplified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-way table

A two-way table, also known as a contingency table, is a tool used in statistics to organize and summarize categorical data. It helps to analyze the relationship between two different variables. For students analyzing survey data, a two-way table is essential for breaking down responses based on categories like gender or habits, such as eating breakfast regularly.

In the provided example, the two-way table categorizes students into groups based on whether they eat breakfast regularly and their gender (male or female). Each cell in the table represents the count of students that fall into a specific category. This makes it very intuitive to immediately grasp how different groups compare in terms of specific behaviors, helping visualize raw data effectively.

Survey analysis

Survey analysis involves collecting, reviewing, and interpreting data obtained from surveys to draw conclusions about a particular subject. When it comes to analyzing survey data, understanding the structure of the questions is crucial, as it determines how the data is tabulated.

In the school breakfast example, the survey asked students if they eat breakfast regularly. With survey analysis, you can:

• Identify patterns (e.g., how many students skip breakfast more often).
• Compare and contrast different groups (e.g., comparing males and females).
• Derive probabilistic statements (e.g., the likelihood of a student eating breakfast).

By strategically organizing survey data into a two-way table, analysts can easily perform survey analysis, making it an engaging way to comprehend the preferences and habits within a population.

Statistical calculations

Statistical calculations involve using mathematical methods to calculate probabilities and other statistical measures. In probability, particularly when dealing with survey data, these calculations can determine the likelihood of a particular event occurring.

In the student survey about breakfast habits, calculations such as finding the probability of selecting a student who is female, or one who eats breakfast regularly, are performed using the formula: \[ P( ext{Event}) = \frac{ ext{Number of favorable outcomes}}{ ext{Total number of possible outcomes}} \]

Some other key concepts in statistical calculations include:

• Simple probabilities: e.g., probability of a female student, calculated as \( \frac{275}{595} \).
• Joint probabilities: e.g., probability of a female who eats breakfast regularly, calculated as \( \frac{110}{595} \).
• Union probabilities involving combinations using the union formula.

Such straightforward mathematical approaches help in in-depth analysis of behavioral patterns.

Student survey data

Student survey data refers to the information collected from surveys conducted among students. This data can cover various topics depending on the objectives of the survey—ranging from habits, preferences, to opinions on certain matters.

For example, the survey conducted by students in an urban school investigating breakfast habits is a kind of primary data collection. Such data is directly obtained from its source, providing authentic insights into students' daily routines.

• It serves as a robust foundation for practical statistical exercises.
• Helps in understanding demographic trends among students.
• Enables assessments which represent the student population's behavior.

This type of data, when properly organized and analyzed, can inform decisions, enhance educational approaches, and even guide further research in school settings.