Assume the machine shifts and is filling the vials with a mean amount of 9.96 milligrams and a standard deviation of 0.05 milligram. You select five vials and find the mean amount of compound added.

c. Which is more sensitive to a shift of parameters—an individual random selection or a randomly selected sample mean?

Verified step by step guidance

Step 1: Understand the concept of sensitivity to parameter shifts. Sensitivity refers to how easily a change in the mean or standard deviation of a distribution can be detected. Individual random selections and sample means respond differently to shifts in parameters.

Step 2: Recall the Central Limit Theorem. When sampling, the mean of a sample (n > 1) is less variable than individual observations. The standard deviation of the sample mean (called the standard error) is calculated as \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size.

Step 3: Compare the distributions shown in the image. The original distribution of sample means (gray curve) has a mean of 9.8 milligrams and is narrower due to reduced variability (smaller standard error). The distribution when the machine shifts (blue curve) has a mean of 9.96 milligrams and is also narrower. This indicates that sample means are less variable and more sensitive to shifts in the mean compared to individual observations.

Step 4: Interpret the impact of the machine shift. A shift in the mean (from 9.8 to 9.96 milligrams) is more noticeable in the distribution of sample means because the narrower spread makes deviations from the expected mean more pronounced. Individual random selections, on the other hand, have greater variability and are less sensitive to such shifts.

Step 5: Conclude that the sample mean is more sensitive to shifts in parameters. This is because the reduced variability (smaller standard error) allows for easier detection of changes in the mean, as illustrated by the narrower and more distinct distributions in the image.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sampling Distribution of the Mean

The sampling distribution of the mean describes the distribution of sample means obtained from a population. When samples of size n are taken, the means of these samples will form a distribution that is normally distributed, centered around the population mean, with a standard deviation known as the standard error. This concept is crucial for understanding how sample means behave and how they can be used to infer properties about the population.

Standard Deviation and Standard Error

Standard deviation measures the amount of variation or dispersion in a set of values. In the context of sampling, the standard error is the standard deviation of the sampling distribution of the mean, calculated as the population standard deviation divided by the square root of the sample size. This concept helps in understanding how much variability we can expect in sample means and is essential for hypothesis testing and confidence intervals.

Sensitivity to Parameter Shifts

Sensitivity to parameter shifts refers to how responsive a statistical measure is to changes in the underlying population parameters. In this context, a sample mean is generally more sensitive to shifts in the population mean than an individual random selection because it aggregates information from multiple observations, thus reflecting changes more clearly. Understanding this sensitivity is important for evaluating the reliability of sample means in detecting shifts in population characteristics.

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