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Chapter 7: Problem 14

The designer of a garbage truck that lifts roll-out containers must estimate the mean weight the truck will lift at each collection point. A random sample of 325 containers of garbage on current collection routes yielded \(x-=75.3 \mathrm{lb}, \mathrm{s}=12.8 \mathrm{lb}\). Construct a \(99.8 \%\) confidence interval for the mean weight the trucks must lift each time.

Short Answer

Expert verified
The 99.8% confidence interval is [73.10, 77.50].

Step by step solution

01

Identify Given Information

We are provided with a random sample of 325 garbage containers, where the sample mean \( \bar{x} = 75.3 \text{ lb} \) and the sample standard deviation \( s = 12.8 \text{ lb} \). We need to construct a \( 99.8 \% \) confidence interval for the mean weight.
02

Determine the Z-score for Confidence Level

For a \( 99.8\% \) confidence interval, the Z-score corresponds to the critical value from the standard normal distribution. Use a Z-table or calculator to find the Z-score: \( Z_{0.999} = 3.09 \).
03

Apply Confidence Interval Formula

The formula for a confidence interval for the population mean when the population standard deviation is unknown is:\[\bar{x} \pm Z \cdot \frac{s}{\sqrt{n}}\]where \(\bar{x} = 75.3\), \(s = 12.8\), \(n = 325\), and \(Z = 3.09\).
04

Calculate the Margin of Error

Calculate the margin of error using the values:\[ME = 3.09 \cdot \frac{12.8}{\sqrt{325}} \approx 2.20\]
05

Find the Confidence Interval

Apply the margin of error to the sample mean to find the confidence interval:\[CI = 75.3 \pm 2.20\]Thus, the confidence interval is \([73.10, 77.50]\).

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Key Concepts

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

Sample Mean
The sample mean is a fundamental concept in statistics. It's essentially the average of a set of numbers found within a sample group.
Imagine you have a random sample of items, like the 325 garbage containers mentioned in the exercise. Each container's weight adds up to a total, which is then divided by the number of containers to find the sample mean.
This is numerically expressed as \( \bar{x} \), and represents a central value of your data set.
  • In our scenario, the sample mean \( \bar{x} \) is 75.3 pounds.
  • This number serves as an estimate for the average weight of all garbage containers the truck lifts on its route.
Understanding the sample mean helps you gain insights into the nature of your data and form a basis for further analysis.
Sample Standard Deviation
The sample standard deviation gives an idea of how spread out the data is. It tells you how far, on average, each data point is from the sample mean.
In simpler terms, it shows variation or dispersion in your sample data set.
For our garbage containers, **s** is the sample standard deviation, noted as \(s = 12.8 \) pounds.
  • A lower value means the weights are relatively close to the mean.
  • A higher value indicates more variation in weight.
Sample standard deviation is crucial because it influences the width of our confidence interval and helps to create a more precise estimation by considering natural variability in data.
Z-score
A Z-score is a measure in statistics used to describe the position of a raw score in terms of its distance from the mean, measured in standard deviations.
It reflects how unusual or normal a data point is within a given distribution. For confidence intervals, the Z-score identifies how far confidence interval limits are from the sample mean.
In our case, we use a Z-score to determine the range for a specific confidence level, here a 99.8% confidence interval.
  • The calculated Z-score for this level is 3.09.
  • This indicates that the point is 3.09 standard deviations away from the mean.
So, the Z-score is critical in establishing the boundaries of our confidence interval that encompasses almost the entirety of where true population parameters are expected to fall.
Margin of Error
The margin of error is an important concept in statistics, as it helps measure the extent of what an observed sample mean might differ from the true population mean.
It provides a range for how far off predictions or estimates are likely to be.
Mathematically, the margin of error is calculated using the formula: \[ ME = Z \cdot \frac{s}{\sqrt{n}} \] Here:
  • \( Z \) is the Z-score, which in this case is 3.09.
  • \( s \) is the sample standard deviation, noted as 12.8 pounds.
  • \( n \) is the number of samples, equaling 325.
For our garbage truck scenario, the margin of error is approximately 2.20 pounds.
Thus, the margin of error helps us build a confidence interval around our sample mean, giving a more dependable estimate of the population mean.

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Most popular questions from this chapter

A sample of 250 workers aged 16 and older produced an average length of time with the current employer ("job tenure") of 4.4 years with standard deviation 3.8 years. Construct a \(99.9 \%\) confidence interval for the mean job tenure of all workers aged 16 or older.

Wildlife researchers tranquilized and weighed three adult male polar bears. The data (in pounds) are: 926,742,1,109 . Assume the weights of all bears are normally distributed. a. Construct an \(80 \%\) confidence interval for the mean weight of all adult male polar bears using these data. b. Convert the three weights in pounds to weights in kilograms using the conversion \(1 \mathrm{lb}=0.453 \mathrm{~kg}\) (so the first datum changes to \((\mathrm{g} 26)(0.453)-410)\). Use the converted data to construct an \(80 \%\) confidence interval for the mean weight of all adult male polar bears expressed in kilograms. c. Convert your answer in part (a) into kilograms directly and compare it to your answer in (b). This illustrates that if you construct a confidence interval in one system of units you can convert it directly into another system of units without having to convert all the data to the new units.

A random sample is drawn from a population of unknown standard deviation. Construct a \(99 \%\) confidence interval for the population mean based on the information given. a. \(\quad n=49, x-17.1, s=2.1\) b. \(\quad n=169, x-=17.1, s=2.1\)

A government agency was charged by the legislature with estimating the length of time it takes citizens to fill out various forms. Two hundred randomly selected adults were timed as they filled out a particular form. The times required had mean 12.8 minutes with standard deviation 1.7 minutes. Construct a \(90 \%\) confidence interval for the mean time taken for all adults to fill out this form.
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