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The U.S. Census Bureau reports that the estimated mean U.S. married-couple family income is \(\$ 90,835 \pm \$ 101 .\) The Bureau describes the margin of error as providing a \(90 \%\) probability that the interval defined by the estimate minus the margin of error and the estimate plus the margin of error (the lower and upper confidence bounds) contains the true value. a. What is the population and variable of interest? b. What parameter is being estimated? What is its estimated value? c. How is the margin of error related to the maximum error of estimate? d. What value is being reported as the margin of error? e. What level of confidence is being reported? f. Find the confidence interval and state exactly what it represents.

Step by step solution

01

Identify Population and Variable of Interest

The population of interest is all U.S. married-couple families and the variable of interest is their income.

02

Identify Parameter and Its Estimated Value

The parameter being estimated is the mean income of U.S. married-couple families. Its estimated value is \$90,835.

03

Understand Margin of Error

The margin of error is related to the maximum error of estimate because it gives a range around the estimated value within which the true value is likely to lie. In this case, it's \$101.

04

Identify Value of Margin of Error

The value being reported as the margin of error is \$101.

05

Identify Confidence Level

The level of confidence being reported is 90%.

06

Find Confidence Interval

The confidence interval is the range from the estimated value minus the margin of error to the estimated value plus the margin of error, so in this case, it's \(\$90,835 - \$101, \$90,835 + \$101\), or \(\$90,734, \$90,936\).

07

Interpret Confidence Interval

In this context, the confidence interval represents that there is a 90% probability that the true mean income of U.S. married-couple families is between \$90,734 and \$90,936.

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Key Concepts

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

Population

In statistics, when we talk about the population, we aren't necessarily referring to the total number of people. Instead, we are focusing on a specific group that a study or survey is investigating. In the context of the U.S. Census Bureau's report, the population includes all U.S. married-couple families.
This encompasses every family unit within the United States where two adults are married, regardless of their geographic location, socioeconomic status, or any other demographic feature. By understanding this population, we can better grasp the significance of the data regarding income.

• The population provides a context for gathering and interpreting data.
• Inferences and conclusions drawn from a sample are intended to reflect this larger group.

Knowing the population helps in identifying relevant target groups for policy-making and social programs.

Margin of Error

The margin of error is a critical concept in statistics, especially when estimating a population parameter like the mean income. It provides a range that indicates where the true value of a parameter likely falls. In our case, the margin of error is reported as $101.
This means we are confident that the true average family income isn't just any random number but is within $101 above or below our estimated mean. The smaller the margin of error, the more precise our estimation is.

• A smaller margin increases our confidence in precise estimates.
• It hinges on the variability of the sampled data and the sample size.

A larger sample usually results in a smaller margin of error, signaling greater precision.

Confidence Level

The confidence level is a percentage that indicates how sure we can be about our estimate being within the margin of error. In this example, the confidence level is 90%.
This essentially means that if the study were repeated multiple times, we expect that 90% of the confidence intervals calculated from those samples would contain the true mean income. A higher confidence level implies greater certainty, but usually also results in a wider confidence interval.

• Higher confidence levels lead to broader intervals.
• Choice of confidence level affects both precision and certainty of estimates.

Deciding on a confidence level often involves balancing the need for precision with the need for certainty.

Mean Income Estimation

Estimating the mean income involves calculating the average income for a defined population, in this case, U.S. married-couple families. The estimated mean income reported by the Census Bureau is $90,835.
Mean estimation provides a central value that represents the entire population, offering insights into economic conditions and resource distribution among families.

• The mean income helps analyze economic disparities.
• It assists in gauging average economic well-being.

Conducting such estimations can inform policymakers and researchers about the living standards and financial health of societal groups.