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Chapter 7: Problem 16

A sample of 50 Fortune 500 companies (Fortune, April 14,2003 ) showed 5 were based in New York, 6 in California, 2 in Minnesota, and 1 in Wisconsin. a. Develop an estimate of the proportion of Fortune 500 companies based in New York. b. Develop an estimate of the number of Fortune 500 companies based in Minnesota. c. Develop an estimate of the proportion of Fortune 500 companies that are not based in these four states.

Short Answer

Expert verified
a. 0.1; b. 20; c. 0.72

Step by step solution

01

Determine the Total Sample Size

The problem states that the sample size is 50 Fortune 500 companies.
02

Calculate the Proportion for New York

To find the proportion of companies based in New York, divide the number of New York companies by the total sample size. Thus, the proportion is \( \frac{5}{50} = 0.1 \).
03

Calculate the Estimate Number for Minnesota

For Minnesota, the sample proportion is \( \frac{2}{50} = 0.04 \). To estimate the number of Fortune 500 companies in Minnesota, multiply this proportion by the total number of Fortune 500 companies, 500. Thus, the estimate is \( 0.04 \times 500 = 20 \).
04

Calculate the Proportion Not Based in These Four States

First, find the total number of companies based in the mentioned 4 states: 5 (New York) + 6 (California) + 2 (Minnesota) + 1 (Wisconsin) = 14. Then, subtract this from the total sample size to find those not based in these states: 50 - 14 = 36. Hence, the proportion is \( \frac{36}{50} = 0.72 \).

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Key Concepts

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

Sample Size
Sample size plays a crucial role in conducting data analysis and drawing accurate conclusions. It represents the number of observations or entities selected from the entire population for study. In our original exercise, the sample consists of 50 Fortune 500 companies.

When estimating proportions or making any kind of statistical inference, it's essential to have a well-chosen sample size.
  • A larger sample size generally gives a more accurate reflection of the population.
  • It helps to reduce variability and potential errors in your estimates.
  • The adequacy of the sample size directly affects the reliability and validity of the conclusions drawn.
Selecting an appropriate sample size can sometimes be tricky and requires statistical techniques, as well as practical and contextual knowledge. By ensuring your sample size is adequate, you lay a strong foundation for further statistical analysis.
Data Analysis
Data analysis involves systematically applying statistical and logical techniques to describe and illustrate, condense and recap, and evaluate data. In our exercise, data analysis was crucial to estimate the proportions of companies based in different states.

Here's a brief walkthrough of the data analysis process used:
  • First, the problem identified was how many Fortune 500 companies are based in specific states, and an estimate of the ones not based in these states.
  • Data collected was the number of companies in each of the selected four states (New York, California, Minnesota, Wisconsin).
  • The proportion analysis was performed to understand the distribution of these companies.
  • Simple arithmetic was used to estimate these proportions and apply them to the total Fortune 500 companies to get practical estimates.
Understanding data analysis helps in making sense of raw data by structuring it into purposeful insights.
Statistical Inference
Statistical inference involves using data from a sample to make generalizations about the larger population. In the exercise provided, we drew inferences about all Fortune 500 companies based on our sample of 50.

The following components are key to understanding statistical inference:
  • Proportion Estimation: Using sample proportions to make estimates about the population, like in our problem with estimating company placements across states.
  • Predictive Analysis: This involves making predictions or forecasts about the population parameters based on the sample data analysis, which was done for the state of Minnesota.
  • Confidence and Errors: While making these inferences, care is taken to express the confidence level and possible errors that could arise, although not explicitly stated in the solution.
Grasping the foundations of statistical inference enables one to make informed and reliable conclusions from sample data, enhancing decision-making processes.

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Most popular questions from this chapter

After deducting grants based on need, the average cost to attend the University of Southern California (USC) is \(\$ 27,175\) (U.S. News \& World Report, America 's Best Colleges, \(2009 \text { ed. }) .\) Assume the population standard deviation is \(\$ 7400 .\) Suppose that a random sample of 60 USC students will be taken from this population. a. What is the value of the standard error of the mean? b. What is the probability that the sample mean will be more than \(\$ 27,175 ?\) c. What is the probability that the sample mean will be within \(\$ 1000\) of the population mean? d. How would the probability in part (c) change if the sample size were increased to \(100 ?\)

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The average price of a gallon of unleaded regular gasoline was reported to be \(\$ 2.34\) in northern Kentucky (The Cincinnati Enquirer, January 21,2006 ). Use this price as the population mean, and assume the population standard deviation is \(\$ .20\). a. What is the probability that the mean price for a sample of 30 service stations is within \(\$ .03\) of the population mean? b. What is the probability that the mean price for a sample of 50 service stations is within \(\$ .03\) of the population mean? c. What is the probability that the mean price for a sample of 100 service stations is within \(\$ .03\) of the population mean? d. Which, if any, of the sample sizes in parts (a), (b), and (c) would you recommend to have at least a .95 probability that the sample mean is within \(\$ .03\) of the population mean?

A production process is checked periodically by a quality control inspector. The inspector selects simple random samples of 30 finished products and computes the sample mean product weights \(\bar{x}\). If test results over a long period of time show that \(5 \%\) of the \(\bar{x}\) values are over 2.1 pounds and \(5 \%\) are under 1.9 pounds, what are the mean and the standard deviation for the population of products produced with this process?

The mean annual cost of automobile insurance is \(\$ 939\) (CNBC, February 23,2006 ). Assume that the standard deviation is \(\sigma=\$ 245\) a. What is the probability that a simple random sample of automobile insurance policies will have a sample mean within \(\$ 25\) of the population mean for each of the following sample sizes: \(30,50,100,\) and \(400 ?\) b. What is the advantage of a larger sample size when attempting to estimate the population mean?
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