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Chapter 8: Problem 6

The Google Play app store for smartphones offers hundreds of games to download for free or for a small fee. The ones for which a fee is charged are called paid games. For a random sample of five paid games taken in July 2014 on the Google platform, the following fees were charged: \(\$ 1.09, \$ 4.99, \$ 1.99, \$ 1.99, \$ 2.99 .\) a. Find a point estimate of the mean fee for paid games available on Google's platform. b. The margin of error at the \(95 \%\) confidence level for this point estimate is \(\$ 1.85 .\) Explain what this means.

Short Answer

Expert verified
Point estimate: \$2.61. Margin of error implies true mean lies within \$1.85 of point estimate.

Step by step solution

01

Identify the Data

We are given a random sample of fees for paid games from Google Play. The fees are \\(1.09, \\)4.99, \\(1.99, \\)1.99, and \$2.99.
02

Calculate the Sample Mean

The point estimate of the mean fee is calculated by finding the average of the given fees. Add up all the fees and then divide by the number of observations: \[\text{Mean} = \frac{1.09 + 4.99 + 1.99 + 1.99 + 2.99}{5}\]Calculate this value to find the mean.
03

Compute the Sample Mean

Perform the arithmetic to find the sample mean:\[\text{Mean} = \frac{13.05}{5} = 2.61\]So, the point estimate of the mean fee is \$2.61.
04

Interpret the Margin of Error

The margin of error for the point estimate, given as \\(1.85 at the 95% confidence level, means that the true mean fee for all paid games on Google's platform is estimated to lie within \\)1.85 above or below our point estimate of \$2.61.

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Key Concepts

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

Point Estimate
A point estimate is a single value that serves as a best guess or approximation of a population parameter, such as the mean. In the context of the exercise, the point estimate is the average fee for paid games on Google Play based on the random sample given.
  • The point estimate provides us with a snapshot value of the mean fee for these games.
  • It is calculated by averaging the sample data.
While it offers a useful estimate, it is not definitive due to the inherent variability present in real-world data. Oftentimes, more statistical measures, such as confidence intervals, are used alongside point estimates to provide a more comprehensive understanding of the likely range of the population parameter.
Sample Mean
The sample mean is a fundamental statistic used to estimate the population mean. It is simply the arithmetic average of the sample data. Here, it was calculated to find the mean fee for the paid games in the Google Play sample. To determine the sample mean:
  • Add up all the sample values. For the given fees: \(1.09 + 4.99 + 1.99 + 1.99 + 2.99 = 13.05\).
  • Divide the total by the number of observations: \(\frac{13.05}{5} = 2.61\).
This result, \(\$2.61\), becomes our sample mean and point estimate of the population mean. It reflects our best estimate of what the true mean fee might be for all paid Google Play games based on the provided sample. This sample mean is useful but does not account for sampling variability, which is why further analysis, often involving the margin of error and confidence intervals, is needed.
Margin of Error
The margin of error quantifies the uncertainty associated with the point estimate. It tells us how much we can expect our point estimate, such as the sample mean, to vary from the true population mean.
  • In this exercise, the margin of error is \(\\(1.85\).
  • When combined with the sample mean, it indicates that the true mean of all paid game fees lies within a certain range.
For a 95% confidence level, it means we can be 95% confident that the interval from \(2.61 - 1.85\) to \(2.61 + 1.85\) includes the true mean. This means the true mean is likely between \(\\)0.76\) and \(\$4.46\). The margin of error provides a buffer for our estimate, taking into account potential variations in the data.
Confidence Interval
A confidence interval offers a range of values within which we can be reasonably certain the true population parameter, like a mean, lies. It complements the point estimate by adding a margin of error to define this range. When we say a 95% confidence interval, it implies:
  • If we took many samples and calculated the interval for each, 95% of these intervals would contain the true mean.
  • In this case, the confidence interval for the mean fee is calculated as \(2.61 \pm 1.85\).
This interval extends from \(\\(0.76\) to \(\\)4.46\). It provides a clearer picture than the point estimate alone by offering a range wherein the true average fee for all such games on Google Play likely falls. This range helps account for sample variability and gives more context about the reliability and precision of the estimate.

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