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Magazine Chapter 24: Problem 9
A machine makes sticks of butter whose average weight is 4.0 ounces; the SD of the weights is 0.05 ounces. There is no trend or pattern in the data. There are 4 sticks to a package. (a) A package weighs (b) A store buys 100 packages. Estimate the chance that they get 100 pounds of butter, to within 2 ounces.
Short Answer
Expert verified
100 packages weigh 1600 ounces (100 pounds), with 95% probability being within 2 ounces of 100 pounds.
Step by step solution
01
Understanding the Mean Weight of a Package
A single stick of butter has an average weight of 4.0 ounces. Since there are 4 sticks in a package, the mean weight of a package, \( \mu_{\text{package}} \), can be calculated by multiplying the average weight of a stick by the number of sticks: \( \mu_{\text{package}} = 4 \times 4.0 \text{ ounces} = 16 \text{ ounces} \).
02
Calculating the Standard Deviation of a Package
The standard deviation for the weight of a stick is 0.05 ounces. Because each package contains 4 sticks, the standard deviation of a package, \( \sigma_{\text{package}} \), is calculated as \( \sigma_{\text{package}} = \sqrt{4} \times 0.05 = 0.1 \text{ ounces} \).
03
Understanding the Weight of 100 Packages
Since a single package weighs an average of 16 ounces, 100 packages weigh an average of \( 100 \times 16 \text{ ounces} = 1600 \text{ ounces} \). Converting this to pounds (since there are 16 ounces in a pound), the average is \( \frac{1600}{16} = 100 \text{ pounds} \).
04
Calculating the Standard Deviation of 100 Packages
Using the standard deviation of a single package (0.1 ounces), the standard deviation for 100 packages, \( \sigma_{100} \), is calculated as \( \sigma_{100} = \sqrt{100} \times 0.1 = 1 \text{ ounce} \).
05
Estimating the Probability for Total Weight
We are asked to find the chance that the total weight of 100 packages is within 2 ounces of 100 pounds (i.e., between 1598 and 1602 ounces). First, convert these bounds into z-scores using the formula: \( z = \frac{X - \mu}{\sigma} \)where \(X\) is the value (1598 and 1602), \(\mu\) is the mean (1600), and \(\sigma\) is the standard deviation (1): - For 1598: \(z = \frac{1598 - 1600}{1} = -2\) - For 1602: \(z = \frac{1602 - 1600}{1} = 2\) The probability that \(z\) is between -2 and 2 is found using the standard normal distribution table, which gives approximately 95% probability.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental concept in statistics that influences how we understand samples and distributions. It states that when we take a sufficiently large number of independent and identically distributed random variables, their sum tends toward a normal distribution, regardless of the original distribution.
In simpler terms, if you repeatedly sample data points and calculate their average, those averages will begin to form a normal distribution, even if the data themselves are not normally distributed. This theorem is crucial for dealing with real-world data that may not neatly fit into a normal shape.
In simpler terms, if you repeatedly sample data points and calculate their average, those averages will begin to form a normal distribution, even if the data themselves are not normally distributed. This theorem is crucial for dealing with real-world data that may not neatly fit into a normal shape.
- With the CLT, the averages become more predictable as sample size increases.
- This predictability is why the theorem is pivotal when calculating probabilities around sample means, like the average weight of butter sticks in a package or the total weight of multiple packages.
Normal Distribution
A normal distribution, often referred to as a bell curve, is a probability distribution that is symmetric about the mean. The symmetry suggests that data near the mean are more frequent in occurrence than data far from the mean.
Normal distributions are characterized by:
In the context of the butter stick problem, once the Central Limit Theorem ensures that the sample means follow a normal distribution, it allows us to use this knowledge to make predictions.
For instance, knowing the normal distribution of the 100 butter package weights helps in determining the likelihood of different total weights when buying in bulk.
Normal distributions are characterized by:
- A mean (\(\mu\)), which determines the midpoint of the distribution.
- A standard deviation (\(\sigma\)), which measures the spread or dispersion of the distribution.
In the context of the butter stick problem, once the Central Limit Theorem ensures that the sample means follow a normal distribution, it allows us to use this knowledge to make predictions.
For instance, knowing the normal distribution of the 100 butter package weights helps in determining the likelihood of different total weights when buying in bulk.
Standard Deviation
Standard deviation is a measure of how spread out numbers in a data set are. It shows the degree of variation from the mean, representing the average amount each data point differs from the mean.
Calculating standard deviation is critical, as it influences how we interpret the variability in data, such as:
The downward scaling of uncertainty with larger sample sizes is explained by this concept, allowing more confidence in predictions for the average of 100 packages.
Calculating standard deviation is critical, as it influences how we interpret the variability in data, such as:
- For one stick of butter with a standard deviation of 0.05 ounces, each stick is typically close to the average weight.
- For four sticks in a package, the overall standard deviation becomes \(\sigma_{\text{package}} = \sqrt{4} \times 0.05 = 0.1 \text{ ounces}\).
- For 100 packages, further spreading the sample size, \(\sigma_{100} = \sqrt{100} \times 0.1 = 1 \text{ ounce}\), indicating variability around the total weight of these packages.
The downward scaling of uncertainty with larger sample sizes is explained by this concept, allowing more confidence in predictions for the average of 100 packages.
Probability Calculation
Calculating probability involves using the normal distribution and standard deviation to understand the likelihood of an event occurring. Here, for the butter packages, the goal is to estimate the probability that their total weight is within a certain range of the expected mean.
Here's how you approach it:
Through these calculations, we apply the principles of the normal distribution and probability theory to make almost certain predictions about large-scale averages.
Here's how you approach it:
- First, determine the mean total weight for 100 packages as 1600 ounces.
- Next, identify the standard deviation of these packages as 1 ounce.
- Convert weight limits (e.g., 1598 and 1602 ounces) into z-scores to standardize these under the normal distribution using \(z = \frac{X - \mu}{\sigma}\).
- The result gives z-scores of -2 and 2.
- Z-scores help find probabilities from the standard normal distribution table, yielding a 95% chance that the total weight falls between 1598 and 1602 ounces.
Through these calculations, we apply the principles of the normal distribution and probability theory to make almost certain predictions about large-scale averages.
