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To investigate how often we eat at home as a family during the week, Harris Interactive surveyed 496 adults living with children under the age of \(18(U S A \text { Today, January } 3,2007\) ). The survey results are shown in the following table. $$\begin{array}{cc}\text { Number of } & \text { Number of Times } \\ \text { Family Meals } & \text { Outcome Occurred } \\ 0 & 11 \\ 1 & 11 \\ 2 & 30 \\\ 3 & 36 \\ 4 & 36 \\ 5 & 119 \\ 6 & 114 \\ 7 \text { or more } & 139\end{array}$$ For a randomly selected family with children under the age of \(18,\) compute the following: a. The probability the family eats no meals at home during the week. b. The probability the family eats at least four meals at home during the week. c. The probability the family eats two or fewer meals at home during the week.

Short Answer

Expert verified
a) 0.0222, b) 0.8226, c) 0.1048

Step by step solution

01

Calculate Total Outcomes

First, we need to find the total number of survey responses. To do this, sum the number of times each outcome occurred. \[11 + 11 + 30 + 36 + 36 + 119 + 114 + 139 = 496\]This confirms the total responses are 496, which matches the number of adults surveyed.
02

Calculate Probability of 0 Meals

The probability that a family eats no meals at home is the ratio of the families reporting 0 meals to the total surveyed.\[P(X=0) = \frac{11}{496}\]\[P(X=0) = 0.0222\]
03

Calculate Probability of At Least 4 Meals

To find the probability of eating at least 4 meals, add the occurrences for 4, 5, 6, and 7 or more meals, then divide by the total. \[P(X \geq 4) = \frac{36 + 119 + 114 + 139}{496}\]\[P(X \geq 4) = \frac{408}{496} = 0.8226\]
04

Calculate Probability of 2 or Fewer Meals

To find the probability of 2 or fewer meals, add the occurrences for 0, 1, and 2 meals, then divide by the total. \[P(X \leq 2) = \frac{11 + 11 + 30}{496}\]\[P(X \leq 2) = \frac{52}{496} = 0.1048\]

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Key Concepts

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

Survey Data Analysis
When we want to understand patterns or behaviors within a population, such as the frequency of family meals at home, surveys become a very useful tool. Surveying involves collecting data from a specific group of people to analyze their responses. Here, Harris Interactive took a thoughtful approach by targeting a sample of 496 adults living with children under 18. This focus helps in understanding family meal dynamics with children involved.
By using survey data, we can analyze the frequency of specific outcomes, like how often a family eats together at home. Once data is gathered, the next step is analysis, where each response is tallied and categorized to draw meaningful insights. This process helps uncover trends or common patterns.
In our example, outcomes are categorized by the number of meals shared at home, with counts for each scenario, helping us observe how frequently such events occur in the sampled families. With the tallied responses, we perform further analysis using statistical calculations.
Family Meal Frequency
Family meal frequency analysis reveals how often families choose to eat together at home during a week. These meals can have a significant impact on family well-being and dynamics. To make such an analysis, the data is divided into groups based on the number of supposed family meals per week. The categories range from 0 up to 7 or more meals.
Understanding these numbers helps us see broader trends, like the most common meal frequency or how often families eat together. This is immensely useful for understanding family eating patterns, which can be important for health and social reasons.
  • "0 meals" might indicate busy schedules or alternate living arrangements.
  • "7 or more meals" suggests a high commitment to shared family time at home.
These insights illustrate family priorities and lifestyle choices, revealing how significant shared meals are in terms of family bonding and nutrition.
Statistical Calculation
Statistical calculations allow us to quantify the likelihood of specific outcomes occurring in our survey. For instance, calculating probabilities helps in understanding potential behaviors of families regarding shared meals.
In this analysis, the formula for probability is straightforward. It's the number of specific outcome occurrences divided by the total number of surveyed families.
Let's break down a few scenarios:
  • **Probability of eating 0 meals**: With 11 families reporting this, the calculation is done by dividing 11 by 496, leading to a probability of 0.0222.
  • **Probability of having at least 4 meals**: Here, we sum totals for 4, 5, 6, and 7+ meals, getting 408 as the total. The probability is then 408 divided by 496, equaling approximately 0.8226.
  • **Probability of 2 or fewer meals**: By adding outcomes for 0, 1, and 2 meals, which sums to 52, we divide by 496. This results in a probability of approximately 0.1048.
These statistical methods provide a clearer picture of family dining habits and their relative likelihoods.

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Most popular questions from this chapter

The Powerball lottery is played twice each week in 28 states, the Virgin Islands, and the District of Columbia. To play Powerball a participant must purchase a ticket and then select five numbers from the digits 1 through 55 and a Powerball number from the digits 1 through 42\. To determine the winning numbers for each game, lottery officials draw five white balls out of a drum with 55 white balls, and one red ball out of a drum with 42 red balls. To win the jackpot, a participant's numbers must match the numbers on the five white balls in any order and the number on the red Powerball. Eight coworkers at the ConAgra Foods plant in Lincoln, Nebraska, claimed the record \(\$ 365\) million jackpot on February \(18,2006,\) by matching the numbers \(15-17-43-44-49\) and the Powerball number \(29 .\) A variety of other cash prizes are awarded each time the game is played. For instance, a prize of \(\$ 200,000\) is paid if the participant's five numbers match the numbers on the five white balls (http://www.powerball.com, March 19, 2006). a. Compute the number of ways the first five numbers can be selected. b. What is the probability of winning a prize of \(\$ 200,000\) by matching the numbers on the five white balls? c. What is the probability of winning the Powerball jackpot?

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