91Ó°ÊÓ

In a Nielsen global online survey of about 27,000 people from 53 different countries, \(61 \%\) of consumers indicated that they purchased more store brands during the economic downturn and \(91 \%\) indicated that they will continue to purchase the same number of store brands when the economy improves. The results were remarkably consistent across all regions of the world. \({ }^{13}\) The survey was conducted during September 2010 and had quotas for age and gender for each country, and the results are weighted to be representative of consumers with Internet access. (a) What is the sample? What is an appropriate population? (b) Find and interpret a \(99 \%\) confidence interval for the proportion of consumers who purchased more store brands during the economic downturn. (c) Find and interpret a \(99 \%\) confidence interval for the proportion of consumers who will continue to purchase the same number of store brands.

Step by step solution

01

Definition of Sample and Population

The sample is a group of about 27,000 people who participated in the Nielsen global online survey. This sample is composed of individuals from 53 different countries. The appropriate population would be all consumers with internet access across the globe, as the sample results are weighted to be representative of this group.

02

Calculation of Confidence Interval for Proportion of Consumers Who Purchased More Brands

A 99% confidence interval for the proportion can be calculated using the formula \[ \hat{p} \pm Z*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] where \(\hat{p}\) is the sample proportion, \(Z\) is the critical value for the 99% confidence level, and \(n\) is the sample size. For this case, \(\hat{p} = 0.61\), \(Z = 2.58\) for a 99% confidence level, and \(n = 27000\). Then substitute these values into the formula to get the confidence interval.

03

Interpretation of Confidence Interval for Proportion of Consumers Who Purchased More Brands

The calculated confidence interval represents the range in which we are 99% confident that the true proportion of consumers who purchased more store brands during the economic downturn falls.

04

Calculation of Confidence Interval for Proportion of Consumers Who Will Continue to Purchase the Same Number of Store Brands

Similar to step 2, the 99% confidence interval can be calculated with \(\hat{p} = 0.91\), \(Z = 2.58\), and \(n = 27000\). Then substitute these values into the formula to get the confidence interval.

05

Interpretation of Confidence Interval for Proportion of Consumers Who Will Continue to Purchase the Same Number of Store Brands

The calculated confidence interval represents the range in which we are 99% confident that the true proportion of consumers who will continue to purchase the same number of store brands when the economy improves falls.

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Key Concepts

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

Sample and Population

Understanding the concepts of sample and population is fundamental in statistics. A sample refers to a subset of individuals selected from a larger group, known as the population. In the exercise, the sample consists of approximately 27,000 people from 53 different countries. These participants have been specifically chosen for the Nielsen global online survey.

When exploring surveys and studies, it's crucial to identify whether the sample is representative of the population. Here, the population is all consumers with internet access worldwide. This distinction is vital as results are usually intended to be generalized beyond the sample to apply to the broader population.
A key element in this context is ensuring the sample reflects the population's characteristics, achieved through strategies such as random sampling and weighting the results as done in this survey.

Proportion

Proportion in statistics indicates the part of the population that exhibits a particular characteristic. It's often expressed as a percentage or fraction. In the given survey, two proportions are of interest:

• 61% of consumers purchased more store brands during the economic downturn.
• 91% of consumers intend to continue purchasing the same amount of store brands after the economy improves.

The calculation of proportions within the context of the survey is handled using the formula for a confidence interval for proportions.
The confidence interval provides a range of plausible values for the population proportion based on the sample data. Hence, it offers a statistical estimate that considers sample variability and the desired confidence level, assisting in making generalizations about the true population.

Statistical Survey

A statistical survey involves collecting data from a subset of a population to glean insights about the whole group. Surveys like the Nielsen global online survey gather extensive data on consumer behavior and preferences.

The design and implementation of a survey are crucial to its success. Important factors include defining clear objectives, selecting an appropriate sample, applying methodological rigor, and analyzing the data reliably. For instance, in this survey, quotas for age and gender were established to ensure the sample accurately represents the population.
Furthermore, weighting techniques adjust for any disparities and enhance the representativeness of the survey results. The thoughtful execution of these strategies ensures the survey results provide a credible, realistic picture of the population's behaviors or opinions.