An air gap exists between fluids in a bottle and the cap of the bottle. At a firm producing olive oil, the gap is critical to the shelf life of their product. If the recent information reveals that the average gap is 9.46 mm with a standard deviation of 0.233, calculate the percentage of product above and below the specification limits of 9.5 +/- 0.5 mm.
The police department of a major city has found that the average height of their 1250 officers is 71 inches with 2.3 inches. How many officers are at least 75 inches tall?
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Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z score. Multiply by 1250.
To calculate the percentage of product above and below the specification limits, we need to first determine the z-scores for the upper and lower limits and then use the standard normal distribution table or a calculator to find the corresponding probabilities.
1. Calculate the z-score for the upper limit:
Z_upper = (Upper Limit - Mean) / Standard Deviation
Z_upper = (9.5 - 9.46) / 0.233
2. Calculate the z-score for the lower limit:
Z_lower = (Lower Limit - Mean) / Standard Deviation
Z_lower = (9.46 - 9.5) / 0.233
3. Find the probabilities associated with the z-scores.
You can use a standard normal distribution table or a calculator to find these probabilities. The area to the left of a z-score represents the percentage of values below that z-score.
4. Calculate the percentage of product below the lower limit:
Percentage_below = Probability(Z < Z_lower) * 100
5. Calculate the percentage of product above the upper limit:
Percentage_above = Probability(Z > Z_upper) * 100
By finding the corresponding probabilities for the z-scores, you can determine the percentage of product above and below the specification limits of 9.5 +/- 0.5 mm.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for ± Z score. Multiply by 100.