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At Farmer's Dairy, a machine is set to fill 32 -ounce milk cartons. However, this machine does not put exactly 32 ounces of milk into each carton; the amount varies slightly from carton to carton. It is known that when the machine is working properly, the mean net weight of these cartons is 32 ounces. The standard deviation of the amounts of milk in all such cartons is always equal to \(.15\) ounce. The quality control department takes a sample of 25 such cartons every week, calculates the mean net weight of these cartons, and makes a \(99 \%\) confidence interval for the population mean. If either the upper limit of this confidence interval is greater than \(32.15\) ounces or the lower limit of this confidence interval is less than \(31.85\) ounces, the machine is stopped and adjusted. A recent sample of 25 such cartons produced a mean net weight of \(31.94\) ounces. Based on this sample, will you conclude that the machine needs an adjustment? Assume that the amounts of milk put in all such cartons have a normal distribution.

Step by step solution

01

Identify the given information

The given information includes the sample mean (\(31.94\) ounces), the population mean (\(32\) ounces), the population standard deviation (\(.15\) ounces), and the sample size (25 cartons). The confidence level is \(99% \).

02

Determine the z-score for the given confidence level

A z-score, which indicates the number of standard deviations an element is from the mean, can be obtained from a standard normal distribution table. For a \(99 %\) confidence level, the z-score is approximately \(2.576\).

03

Calculate the standard error of the mean

The standard error of the mean is the standard deviation divided by the square root of the sample size. In this case, \(.15 / \sqrt{25} = .03\).

04

Calculate the confidence interval

The confidence interval is calculated by subtracting and adding the product of the z-score and the standard error from the sample mean. This gives the interval \(31.94 - 2.576*.03 \) to \(31.94 + 2.576*.03\), which equals approximately \(31.86\) to \(32.02\).

05

Make a conclusion based on the confidence interval

Given the specified machine adjustment range (\(31.85\) to \(32.15\) ounces), the confidence interval falls within the acceptable range. Therefore, there is no need for an adjustment based on this sample.

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Key Concepts

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

Standard Deviation

Standard deviation is a critical concept in statistics that measures the amount of variation or dispersion within a set of data. It tells us how spread out the numbers in a data set are.
In the context of the Farmer's Dairy milk carton filling machine, the standard deviation is given as \(0.15\) ounces. This means that most milk carton weights will fall within \(0.15\) ounces of the 32-ounce mean if the machine is functioning correctly.

The standard deviation helps in understanding how consistent the machine is with filling each carton. If the standard deviation is small, it indicates that the carton weights are very close to the mean. If it’s larger, there’s more variability in the carton weights. This concept is vital to determine if the machine needs adjustments by understanding the consistency of its performance.

Sample Mean

The sample mean is the average of a set of observations and gives us an estimate of the population mean. In statistical studies, it provides insight into the average behavior of a population based on a sample.
For Farmer's Dairy, the sample mean is obtained from 25 milk cartons and is calculated to be \(31.94\) ounces.
This is below the population mean of 32 ounces, which suggests the machine might be slightly underfilling the cartons on average.

The sample mean forms the base of constructing a confidence interval to statistically ascertain whether this deviation is within acceptable levels or not. By calculating the sample mean, we can use it to infer conclusions about the entire population and make decisions regarding machine performance.

Z-Score

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean. For example, a z-score of \(2.576\) indicates that the value is 2.576 standard deviations away from the mean.

In the Farmer's Dairy problem, the z-score is used to establish the \(99\%\) confidence interval. This confidence interval helps determine if the current machine operation is within acceptable standard deviations from the desired mean.

• The z-score for a \(99\%\) confidence level corresponds to approximately \(2.576\), as it covers almost all of the expected data in a normal distribution.

Employing the z-score, the dairy can figure out how likely it is that the observed sample mean of \(31.94\) ounces is a result of random sampling variation or if it demands corrective measures.

Normal Distribution

The normal distribution is a bell-shaped probability distribution central to statistics. It describes how the values of a variable are distributed, showing that most occurrences take place near the central peak and probabilities for values further away from the mean taper off symmetrically in both directions.

In the given exercise, the amounts of milk filled in the cartons are assumed to follow a normal distribution. This assumption is critical to accurately applying statistical methods like confidence intervals and z-scores.

• It means that while some cartons might be slightly underfilled or overfilled, the majority will cluster around the mean of 32 ounces.
• Because of the characteristics of a normal distribution, we can use the corresponding properties and tables, like the z-table, to make statistical inferences about the population based on sample statistics.

Understanding normal distribution allows the quality control department to make informed decisions about when to adjust the machine, ensuring optimal performance and product quality.