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91Ó°ÊÓ

According to Nielsen/ NetRatings, the mean amount of time spent on MySpace.com per user per month in July 2014 was 171.0 minutes. A \(95 \%\) confidence interval for the mean amount of time spent on MySpace.com monthly has a lower bound of 151.4 minutes and an upper bound of 190.6 minutes. What does this mean?

Short Answer

Expert verified
We are 95% confident that the true mean amount of time users spent on MySpace.com per month in July 2014 is between 151.4 and 190.6 minutes.

Step by step solution

01

Understand Confidence Interval

A confidence interval is a range of values that is likely to contain the population mean. Here, a 95% confidence interval means we are 95% confident that the true mean amount of time spent by users on MySpace.com falls within this interval.
02

Identify the Given Data

The given data includes a mean amount of time spent on MySpace.com per user per month, which is 171.0 minutes. The lower bound of the confidence interval is 151.4 minutes, and the upper bound is 190.6 minutes.
03

Interpret the Confidence Interval

The statement means that we can be 95% confident that the true mean amount of time users spend on MySpace.com per month lies between 151.4 minutes and 190.6 minutes. In other words, if we took many samples and calculated the confidence interval each time, approximately 95% of the intervals would contain the true mean.

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Key Concepts

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

Mean
The mean is a fundamental concept in statistics. It represents the average value of a set of numbers. To calculate the mean, you sum up all the values in a dataset and divide by the number of values. For example, if you have five numbers: 5, 10, 15, 20, and 25, their mean is \((5+10+15+20+25)/5 = 15\). The mean provides a central measure of the dataset.
In the context of the exercise, the mean amount of time spent on MySpace.com per user per month is 171.0 minutes. This figure helps us understand how much time, on average, users spend on the platform monthly.
Population Mean
The population mean is the average value of a particular measure across an entire population. Unlike the sample mean, which is calculated from a subset of the population, the population mean encompasses the entire group's data.
In our exercise, the population mean refers to the average time every user, not just those in a sample, spends on MySpace.com per month. When we're given a population mean of 171.0 minutes, it means that if we could measure the time for every user, the average would be 171.0 minutes.
Knowing the population mean helps us make informed decisions and better understand the behaviors or characteristics of the whole group.
Interval Estimation
Interval estimation is a range within which a population parameter lies with a certain level of confidence. Unlike point estimation, which provides a single value estimate, interval estimation gives an upper and lower bound for the estimate.
For example, if we want to estimate the population mean with an interval, we calculate a range around the sample mean. In the exercise, we're given a 95% confidence interval ranging from 151.4 minutes to 190.6 minutes. This means we are 95% confident that the true mean falls within this range.
Interval estimation allows us to understand the possible variation and provides a more comprehensive picture of the data.
95% Confidence Level
A 95% confidence level means that if we were to take 100 different samples and calculate a confidence interval for each sample, approximately 95 of those intervals would contain the true mean. This does not guarantee that any one specific interval will contain the true mean, but it does give us a high level of confidence.
In the exercise, when we say there's a 95% confidence interval from 151.4 minutes to 190.6 minutes, it means we are 95% confident that the true mean time spent on MySpace.com falls within this range. This high level of confidence helps reinforce our findings and lends credibility to our statistical analysis.
Understanding the 95% confidence level ensures that our estimates are reliable and shows the precision of our measurement.

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