6. A manufacturer regularly receives shipments of computer chips. Over the past year the mean number of computer chips per shipment was 1200. If a shipment of 1456 chips was 1.6 standard deviations above the mean how many standard deviations below the mean was a shipment of 848 chips?
Answer: E
A shipment of 848 chips was 2.2 standard deviations below the mean.
To determine how many standard deviations below the mean a shipment of 848 chips is, we first need to find the standard deviation. Given that a shipment of 1456 chips is 1.6 standard deviations above the mean of 1200, we can calculate the standard deviation as follows:
1. Calculate the difference between 1456 and the mean (1200):
1456 - 1200 = 256.
2. Since this difference represents 1.6 standard deviations:
Standard deviation (σ) = 256 / 1.6 = 160.
3. Now, we find how many standard deviations below the mean the shipment of 848 chips is:
1200 - 848 = 352.
4. Divide this difference by the standard deviation:
352 / 160 = 2.2.
A) 1.3
This option suggests that the shipment of 848 chips is 1.3 standard deviations below the mean. However, the calculations indicate that it is actually 2.2 standard deviations below the mean, making this option incorrect.
B) 1.4
This choice posits that the shipment of 848 chips is 1.4 standard deviations below the mean. Similar to option A, this is inaccurate based on the calculated value of 2.2 standard deviations below the mean.
C) 1.7
Option C states that the shipment of 848 chips is 1.7 standard deviations below the mean. This is also incorrect, as our calculations show a value that is higher than 1.7, specifically 2.2 standard deviations below the mean.
D) 2
This option suggests that the shipment of 848 chips is 2 standard deviations below the mean. While this is closer to the correct answer, it still does not account for the full deviation, which is 2.2 standard deviations below the mean.
E) 2.2
This option accurately reflects the calculation of how many standard deviations the shipment of 848 chips is below the mean. It is derived from the difference between the mean and the shipment value, divided by the standard deviation, leading to a definitive answer of 2.2 standard deviations below the mean.
Conclusion
The correct answer of 2.2 standard deviations below the mean is supported by the calculations based on the provided data. All other options fail to represent the true standard deviation distance, either underestimating or inaccurately stating the value, thus confirming that option E is the only correct choice in this context.