Large Sample Normal Approximation in Quality Control

00:00 6 marks

A factory manufactures electronic components, and the weights of these components are approximately normally distributed with a population mean (\(\mu\)) of 50 grams and a population standard deviation (\(\sigma\)) of 2 grams. For quality control purposes, a manager randomly selects a sample of 100 components and calculates the sample mean weight, \(\bar{x}\).

Calculate the probability that the sample mean weight \(\bar{x}\) lies between 49.8 grams and 50.2 grams. Interpret your result in the context of quality control.

Hint Hint

Hint

To solve this, first calculate the standard error of the sample mean, then standardize the interval bounds using z-scores, and finally find the probability using the standard normal distribution. Remember to interpret what this probability means for the quality control process. Consider using the normal cumulative distribution function (normalcdf) on your GDC.

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Answer Answer

Complete Answer

Approximately 0.683 or 68.3%. This means there is a 68.3% chance that a random sample of 100 components will have a mean weight between 49.8 and 50.2 grams.

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Answer Explanation Answer Explanation

Examiner Tips Examiner Tips

Common Errors Common Errors

Focus Focus

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