A biscuit manufacturer packs its biscuits into individual boxes using a machine. The weight is advertised on each box as 200 grammes. The machine has been operating for many years and is regularly inspected during production to ensure that it is operating correctly. A quality control worker at the factory has used historical data from many previous production runs. It is known that the weight of packed boxes follows a normal distribution with a mean weight of 200 grammes and a standard deviation of 2.1 grammes. During a particular production run, a quality-control worker randomly selects one box and finds that the weight is 208.1 grammes.

Question

A biscuit manufacturer packs its biscuits into individual boxes using a machine. The weight is advertised on each box as 200 grammes. The machine has been operating for many years and is regularly inspected during production to ensure that it is operating correctly. A quality control worker at the factory has used historical data from many previous production runs. It is known that the weight of packed boxes follows a normal distribution with a mean weight of 200 grammes and a standard deviation of 2.1 grammes. During a particular production run, a quality-control worker randomly selects one box and finds that the weight is 208.1 grammes.

Required
Using the empirical rule and the data from the machine’s historical operational performance, determine whether this box weight of 208.1 grammes would be considered usual or unusual.

Answer 1

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Answer 2

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Answer 3

To determine whether the box weight of 208.1 grams would be considered usual or unusual, we can use the empirical rule, also known as the 68-95-99.7 rule.

According to the empirical rule:
- Approximately 68% of the data lies within one standard deviation of the mean.
- Approximately 95% of the data lies within two standard deviations of the mean.
- Approximately 99.7% of the data lies within three standard deviations of the mean.

Given that the mean weight of the packed boxes is 200 grams and the standard deviation is 2.1 grams, we can calculate the boundaries for what would be considered usual or unusual.

1) Calculate the lower boundary for the usual range:
Lower Boundary = Mean - (1 * Standard Deviation)
= 200 - (1 * 2.1)
= 197.9 grams

2) Calculate the upper boundary for the usual range:
Upper Boundary = Mean + (1 * Standard Deviation)
= 200 + (1 * 2.1)
= 202.1 grams

Therefore, any box weight between 197.9 grams and 202.1 grams would be considered usual based on the historical data.

Since the weight of the selected box is 208.1 grams, which is above the upper boundary of 202.1 grams, it would be considered unusual based on the empirical rule and the data from the machine's historical operational performance.