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York Steel Corporation produces iron rings that are supplied to other companies. These rings are supposed to have a diameter of 24 inches. The machine that makes these rings does not produce each ring with a diameter of exactly 24 inches. The diameter of each of the rings varies slightly. It is known that when the machine is working properly, the rings made on this machine have a mean diameter of 24 inches. The standard deviation of the diameters of all rings produced on this machine is always equal to \(.06\) inch. The quality control department takes a sample of 25 such rings every week, calculates the mean of the diameters for these rings, and makes a \(99 \%\) confidence interval for the population mean. If either the lower limit of this confidence interval is less than \(23.975\) inches or the upper limit of this confidence interval is greater than \(24.025\) inches, the machine is stopped and adjusted. A recent such sample of 25 rings produced a mean diameter of \(24.015\) inches. Based on this sample, can you conclude that the machine needs an adjustment? Explain. Assume that the population distribution is normal.

Step by step solution

01

Calculate Standard Error

First we need to calculate the standard error (SE) of the mean. The standard error is a measure of how spread out the means of the sample distribution are around the population mean. We calculate it using the formula: SE = \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the standard deviation and \(n\) is the sample size. Here, \(\sigma = .06\) and \(n = 25\). Thus, SE = \(\frac{.06}{\sqrt{25}} = .012\)

02

Calculate Confidence Interval

Next, we calculate the 99% confidence interval for the population mean. The confidence interval is given by the formula: \(\bar{x} \pm Z*SE\), where \(\bar{x}\) is the sample mean, \(Z\) is the Z-score corresponding to the given percentage (for a 99% confidence interval, \(Z = 2.576\)), and SE is the calculated standard error. Here, \(\bar{x} = 24.015\) and SE = 0.012. Hence the confidence interval is \(24.015 \pm 2.576 * .012\)

03

Determine if the Machine Needs Adjustment

Finally, determine if the calculated confidence interval falls within the desired range of \(23.975 - 24.025\). If the confidence interval falls outside this range, the machine will need adjustment.

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Key Concepts

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

Standard Error

The Standard Error (SE) is a fundamental concept in statistics that helps us understand how sample means compare to the population mean. In simpler terms, it tells us how much sample means deviate from what we expect them to be. This is essential when taking repeated samples from a population.

To calculate SE, we use the formula:

• SE = \(\frac{\sigma}{\sqrt{n}}\)
• \(\sigma\) represents the standard deviation of the population.
• \(n\) is the number of observations or sample size.

By dividing the population standard deviation by the square root of the sample size, we adjust for the fact that larger samples tend to be more representative of the entire population and thus, their means exhibit less variability. In our example, with a population standard deviation of \(0.06\) and a sample size of \(25\), we find the standard error is \(0.012\). This means there's relatively little variation in our sample means, which provides a more accurate estimate of the true population mean.

Z-score

The Z-score is a statistical measure that helps to establish how far away a given data point is from the mean, expressed in terms of standard deviations. It is widely used when constructing confidence intervals or testing hypotheses.

In this exercise, we use the Z-score to determine the width of the confidence interval around the sample mean. For a 99% confidence level, which is quite stringent, the Z-score is \(2.576\). This means we are 99% confident that the true population mean falls within this range of values.

• A higher Z-score means greater confidence level, leading to a wider confidence interval.
• The Z-score works under the assumption of a normal distribution, which our example satisfies.
• It helps account for variability in sampling, ensuring conclusions are statistically sound.

Therefore, to form a confidence interval, we multiply the Z-score by the standard error (SE). This gives us the margin of error which helps in understanding the precision of our estimation.

Population Mean

The Population Mean is a critical concept in statistics, representing the average of a set of characteristics in an entire population. In practical terms, it indicates the center or "central tendency" of the data.

In our scenario, the population mean is noted to be exactly 24 inches, which is the target diameter for the iron rings. Understanding the population mean allows us to gauge the extent by which the sample mean differs, and whether this deviation is statistically significant.

• The sample mean is used to estimate the population mean, especially when measuring every single item in a population isn’t feasible.
• If sample means are consistently and significantly different from the population mean, it may signal issues like machine tampering or errors.
• While sample mean offers a view into the likely population mean, confidence intervals help define this estimate's accuracy.

Hence, keeping the population mean in sight is essential to ensure that the process, like ring manufacturing, remains consistent and within expected boundaries.

Quality Control

Quality Control is an overarching term for the methods and processes used to ensure products meet certain standards and specifications. It is crucial in manufacturing settings to ensure consistency and detect anomalies in production processes.

In this scenario, quality control procedures are applied by calculating confidence intervals to regularly check whether the machine producing the rings needs adjustment. If the confidence interval falls outside the specified range (23.975 to 24.025 inches), corrective measures are taken.

• It continually monitors and evaluates the production output.
• Quality control ensures that products do not deviate from the intended design or dimensions, maintaining the brand's or company's reputation for reliability.
• It significantly minimizes waste and the costs associated with reworking defective products.

Using statistical tools like Z-scores and confidence intervals, quality control departments can effectively maintain optimal operational standards and quickly react to any discrepancies in the production process.