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In a BusinessWeek/Harris Poll, 1035 adults were asked about their attitudes toward business (BusinessWeek, September 11, 2000). One question asked: "How would you rate large U.S. companies on making good products and competing in a global environment?" The responses were: excellent- \(18 \%\), pretty good \(-50 \%\), only fair- \(26 \%\), poor \(-5 \%\) and don't know/no answer- \(1 \%\) a. What is the probability that a respondent rated U.S. companies pretty good or excellent? b. How many respondents rated U.S. companies poor? c. How many respondents did not know or did not answer?

Short Answer

Expert verified
a. 68%; b. 52; c. 10 respondents

Step by step solution

01

Identify Probability Components

We are given percentages for different responses in a survey. To find the probability that a respondent rated U.S. companies as "pretty good" or "excellent," we add those two percentages together. The probability can be calculated as the sum of the percentages divided by 100.
02

Calculate Total Probability for 'Pretty Good' or 'Excellent'

Since 18% of respondents rated companies as 'excellent' and 50% as 'pretty good', add these percentages: \[18\% + 50\% = 68\%\]. So, the probability is 68%.
03

Determine Number of 'Poor' Ratings

To find out how many respondents rated the companies as 'poor,' multiply the total number of respondents by the percentage who rated them as 'poor' (5%): \[1035 \times \frac{5}{100} = 51.75\].Since the number of respondents must be a whole number, round to 52.
04

Determine Number of 'Don't Know/No Answer' Ratings

To find the number of respondents who did not know or did not answer, similarly multiply the total number of respondents by the percentage (1%): \[1035 \times \frac{1}{100} = 10.35\].Round to the nearest whole number, resulting in 10.

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Key Concepts

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

Survey Analysis
Survey analysis involves systematically gathering and interpreting data collected from a group of respondents to draw conclusions about a larger population. In the context of our exercise, the survey conducted asked participating adults to rate large U.S. companies based on their product quality and competitiveness in a global environment. The responses were quantified, with each category receiving a specific percentage of opinions.
By analyzing the percentages of each response, we can identify trends and overall sentiment towards U.S. companies. For example, combining the percentages of 'pretty good' and 'excellent' responses gives us a comprehensive view of favorable attitudes. Survey analysis often involves these steps to ensure accuracy:
  • Defining survey questions clearly to elicit specific information.
  • Collecting data through structured surveys, ensuring a reliable sample size.
  • Examining response categories to calculate probabilities, such as finding the likelihood of a specific response.
This process allows businesses to tailor their strategies based on consumer perceptions, and use statistical methods to validate hypotheses about public opinion.
Data Interpretation
Data interpretation is an essential part of analyzing survey results. It involves taking the raw percentages and turning them into meaningful insights. For instance, in the provided exercise, we interpret the data by calculating the probability that a respondent will rate U.S. companies as 'pretty good' or 'excellent'.
In this case, the answer involves adding the percentages of these categories together to determine that there is a 68% chance a respondent would rate the companies favorably. Such interpretation requires understanding:
  • The context behind the data, such as the reason for choosing specific percentages instead of raw numbers.
  • How to apply statistical operations to convert these percentages into practical information used for decision-making.
  • The significance of rounding off results when calculating the number of respondents as seen in the steps involving 'poor' and 'don't know/no answer' categories.
Interpreting this data correctly supports businesses in recognizing the areas of improvement and strengths as perceived by the public.
Statistics in Business
Statistics play a crucial role in business decision-making and strategy development. They enable businesses to make sense of vast amounts of survey data and extract valuable insights. By using statistical techniques, businesses can perform tasks such as forecasting trends, assessing risks, and making informed decisions.
The survey results in this exercise show how probabilities and percentages can provide a snapshot of consumer attitudes. The calculation of probabilities tells us about the general sentiment towards U.S. companies, which can be crucial in strategic business planning.
  • Businesses use statistical distribution to understand variability in consumer opinions.
  • Analyzing survey data helps identify market opportunities and competitive advantages.
  • Correct application of statistical methods ensures data-driven and customer-focused business strategies.
Ultimately, statistics in business facilitate measuring outcomes against set objectives, optimizing performance, and driving growth based on data-backed analyses.

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Most popular questions from this chapter

In early \(2003,\) President Bush proposed eliminating the taxation of dividends to shareholders on the grounds that it was double taxation. Corporations pay taxes on the earnings that are later paid out in dividends. In a poll of 671 Americans, TechnoMetrica Market Intelligence found that \(47 \%\) favored the proposal, \(44 \%\) opposed it, and \(9 \%\) were not sure (Investor's Business Daily, January 13,2003 ). In looking at the responses across party lines the poll showed that \(29 \%\) of Democrats were in favor, \(64 \%\) of Republicans were in favor, and \(48 \%\) of Independents were in favor. a. How many of those polled favored elimination of the tax on dividends? b. What is the conditional probability in favor of the proposal given the person polled is a Democrat? c. Is party affiliation independent of whether one is in favor of the proposal? d. If we assume people's responses were consistent with their own self- interest, which group do you believe will benefit most from passage of the proposal?

Assume that we have two events, \(A\) and \(B\), that are mutually exclusive. Assume further that we know \(P(A)=.30\) and \(P(B)=.40\) a. What is \(P(A \cap B) ?\) b. What is \(P(A | B) ?\) c. A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer. d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

Venture capital can provide a big boost in funds available to companies. According to Venture Economics (Investor's Business Daily, April 28, 2000), of 2374 venture capital disbursements, 1434 were to companies in California, 390 were to companies in Massachusetts, 217 were to companies in New York, and 112 were to companies in Colorado. Twenty-two percent of the companies receiving funds were in the early stages of development and \(55 \%\) of the companies were in an expansion stage. Suppose you want to randomly choose one of these companies to learn about how venture capital funds are used. a. What is the probability the company chosen will be from California? b. What is the probability the company chosen will not be from one of the four states mentioned? c. What is the probability the company will not be in the early stages of development? d. Assume the companies in the early stages of development were evenly distributed across the country. How many Massachusetts companies receiving venture capital funds were in their early stages of development? e. The total amount of funds invested was \(\$ 32.4\) billion. Estimate the amount that went to Colorado.

The prior probabilities for events \(A_{1}\) and \(A_{2}\) are \(P\left(A_{1}\right)=.40\) and \(P\left(A_{2}\right)=.60 .\) It is also known that \(P\left(A_{1} \cap A_{2}\right)=0 .\) Suppose \(P\left(B \text { ? } A_{1}\right)=.20\) and \(P\left(B | A_{2}\right)=.05\) a. Are \(A_{1}\) and \(A_{2}\) mutually exclusive? Explain. b. Compute \(P\left(A_{1} \cap B\right)\) and \(P\left(A_{2} \cap B\right)\) c. Compute \(P(B)\) d. Apply Bayes' theorem to compute \(P\left(A_{1} | B\right)\) and \(P\left(A_{2} | B\right)\).

The prior probabilities for events \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(A_{1}\right)=.20, P\left(A_{2}\right)=.50,\) and \(P\left(A_{3}\right)=\) \(.30 .\) The conditional probabilities of event \(B\) given \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(B | A_{1}\right)=.50\) \(P\left(B | A_{2}\right)=.40,\) and \(P\left(B | A_{3}\right)=.30\) a. Compute \(P\left(B \cap A_{1}\right), P\left(B \cap A_{2}\right),\) and \(P\left(B \cap A_{3}\right)\) b. Apply Bayes' theorem, equation (4.19), to compute the posterior probability \(P\left(A_{2} | B\right)\). c. Use the tabular approach to applying Bayes' theorem to compute \(P\left(A_{1} | B\right), P\left(A_{2} | B\right)\) and \(P\left(A_{3} | B\right)\)

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