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To understand the concept of sampling distribution, imagine a scenario where you take multiple samples from a larger population. In this case, think about repeatedly selecting 500 chips from an endless supply, each time noting the fraction that is defective. The collection of these proportions forms what is called a sampling distribution. Sampling distributions allow us to infer about a population parameter based on sample statistics.
In this exercise, the focus is on the sample proportion of defective chips. Knowing this allows us to make statistical inferences, such as understanding the variability or estimating probabilities.

• The sample proportion of defective chips was found by looking at the proportion of defectives in a single sample of 500 chips.
• The parameter of interest here is the population proportion of defective chips, denoted as \( p \).
• The mean of this sampling distribution gives the average expected proportion, which mirrors the population proportion, \( p = 0.04 \).

It’s important because it lays the foundation for using sample statistics to make statements about the population.

The normal distribution, often depicted as a bell-shaped curve, is a fundamental concept in statistics because many natural phenomena tend to follow its pattern. When you have a large enough sample size, the Central Limit Theorem states that the sample mean will tend to be normally distributed, even if the underlying distribution is not.In this problem, we wish to approximate whether the distribution of our sample proportion follows a normal distribution. To do this, certain conditions must be checked:

• The product \( np \) must be at least 10. This ensures enough data points exist to see the normal curve's shape.
• The product \( n(1-p) \) must also be at least 10, ensuring adequate non-defective data.

Here, both conditions are met (with values of 20 and 480, respectively). Thus, this supports modeling our distribution as normal, allowing for the use of the normal distribution's properties in subsequent calculations. This helps us understand and apply probabilities to our issues effectively.

Standard deviation is a measure that captures the amount of variation or dispersion of a set of values. In the context of a sampling distribution, it helps to understand how much variability there is in the sample proportion as you repeatedly sample from the population.
The formula for the standard deviation of a sample proportion (\( \sigma_p \)) is:\[\sigma_p = \sqrt{\frac{p(1-p)}{n}}\]In this exercise, using the provided proportion \( p = 0.04 \) and \( n = 500 \), we calculated \( \sigma_p \approx 0.0087 \). This relatively small standard deviation indicates that the proportion of defective chips in any sample of 500 chips is likely to be close to the expected proportion (0.04). The small value suggests high consistency across repeated samples.Understanding and correctly applying standard deviation is crucial because it directly impacts confidence intervals and hypothesis tests, influencing decisions based upon sample data.

A Z-score is a metric that expresses how many standard deviations an element is from the mean. This becomes useful when comparing a specific observation from a dataset to the expected normal distribution.In the problem, we convert a sample proportion of defective chips, \( \hat{p} = 0.05 \), to a Z-score to understand the position of this sample proportion within the normal distribution of the population.The formula for calculating a Z-score is:\[Z = \frac{\hat{p} - \mu_p}{\sigma_p}\]By substituting our values (\( \mu_p = 0.04 \) and \( \sigma_p = 0.0087 \)), we found a Z-score of approximately 1.15. This suggests that the sample proportion of 0.05 is 1.15 standard deviations above the mean proportion of defects.Z-scores help us assess probabilities of sample means, compare different datasets, and make decisions. For instance, using Z-scores and a Z-table, the manufacturer can determine the likelihood of rejecting a shipment, helping in quality control and cost management.