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After Rob Ford, the mayor of Toronto, announced his plans to cut budget costs in late 2011, the Forum Research polled 1,046 people to measure the mayor’s popularity. Everyone polled expressed either approval or disapproval. These are the results their poll produced: • In early 2011, 60 percent of the population approved of Mayor Ford’s actions in office. • In mid-2011, 57 percent of the population approved of his actions. • In late 2011, the percentage of popular approval was measured at 42 percent. a. What is the sample size for this study? b. What proportion in the poll disapproved of Mayor Ford, according to the results from late 2011? c. How many people polled responded that they approved of Mayor Ford in late 2011? d. What is the probability that a person supported Mayor Ford, based on the data collected in mid-2011? e. What is the probability that a person supported Mayor Ford, based on the data collected in early 2011?

Step by step solution

01

Determining the Sample Size

The problem states that Forum Research polled 1,046 people to measure the mayor’s popularity. Therefore, the sample size for this study is 1,046.

02

Calculating Late 2011 Disapproval Proportion

In late 2011, 42% of the population approved of Mayor Ford. To find the disapproval proportion, subtract the approval percent from 100%: \[ 100\% - 42\% = 58\% \].Hence, the disapproval proportion is 58%.

03

Calculating Approval Responses in Late 2011

To find the number of people who approved of Mayor Ford in late 2011, use the 42% approval rate:\[ 0.42 \times 1046 = 439.32 \].Since the number of people must be a whole number, round to 439 people.

04

Calculating Mid-2011 Support Probability

The probability that a randomly selected person approved in mid-2011 is 57%. Express this as a decimal:\[ \text{Probability} = 0.57 \].

05

Calculating Early 2011 Support Probability

The probability that a randomly selected person approved in early 2011 is 60%. Express this as a decimal:\[ \text{Probability} = 0.60 \].

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Key Concepts

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

Poll Data Interpretation

Poll data interpretation involves understanding and analyzing survey results to glean insights about public opinion on specific topics. In our example with Mayor Ford, interpreting the poll results requires identifying explicit information such as the time frame of each survey and noting the population percentage that approves or disapproves of his political actions.

This skill helps to understand trends over time, which can highlight shifts in public opinion. By looking at early, mid, and late 2011, we can see approval ratings dropped from 60% to 42%. These changes provide essential insights into how opinions evolved, likely influenced by the mayor's policy decisions or specific events.

• Time-based poll comparison helps highlight shifts.
• Raw data, such as percentage approvals, are essential to spot trends.

Developing this interpretation skill means being able to contextualize numbers, understand percentages, and derive meaning from them. The poll data isn't just numbers but a story of public perception which needs to be fully understood.

Approval and Disapproval Rates

Approval and disapproval rates provide a snapshot of public sentiment at any given point in time. These percentages help gauge the effectiveness of actions taken by a person or entity, represented here by Mayor Ford's administration. In our scenario, an approval rate of 42% in late 2011 also tells us that 58% of the population disapproved, a crucial understanding.

Calculating these rates accurately is important. To find the disapproval rate when given the approval rate, subtract the approval percentage from 100%. This helps balance the view and better understand overall sentiment. For example:

• Approval rate in late 2011 is 42%.
• Disapproval rate = 100% - 42% = 58%.

Knowing how these figures relate tells us much about public perception and allows for informed decision-making. It also emphasizes the significance of majority sentiment, as represented by the higher disapproval rate in this case, indicating a need for strategic responsiveness.

Probability Calculations

Probability calculations allow us to quantify the likelihood of specific outcomes based on certain data. In polling situations, this means calculating the probability that an individual had a specific opinion during a surveyed time. Here, the calculations convert percentage-based data into probabilities.

For example, if 57% of respondents approved of Mayor Ford in mid-2011, we express the probability that a person supported the mayor as 0.57. This conversion is straightforward:

• Convert approval percentage to decimal for probability.
• Example for mid-2011: 57% becomes 0.57.

Probability gives us an easy way to communicate public sentiment in tangible terms, making these figures useful for predictions and strategic planning. This mathematical approach helps simplify raw data interpretation and emphasizes the likelihood, ensuring that decisions can be pragmatic and data-driven.