Question

The owners of the Burger Emporium are looking for new supplier of onions for their famous hamburgers. It is important that the onion slice be roughly the same diameter as the hamburger patty. After careful analysis, they determine that they can only use onions with diameters between 9 and 10 cm. Company A provides onions with diameters that are approximately normally distributed with mean 10.3 cm and standard deviation of 1.2 cm. Company B provides onions with diameters that are approximately normally distributed with mean 10.6 cm and standard deviation of 0.9 cm. Which company provides the higher proportion of usable onions? Justify your choice with an appropriate statistical argument using proportions.

Answer 1

To find the proportion of usable onions for each supplier, calculate the area under the normal distribution curve between the specified diameter range. Company B provides a higher proportion of usable onions compared to Company A.

Step-by-step explanation:

To find the proportion of usable onions for each supplier, we need to calculate the area under the normal distribution curve between the specified diameter range. For Company A, the mean diameter is 10.3 cm with a standard deviation of 1.2 cm. For Company B, the mean diameter is 10.6 cm with a standard deviation of 0.9 cm.

First, we calculate the z-scores for the lower and upper limits of the diameter range for each supplier. For Company A, the z-score for the lower limit (9 cm) is (9 - 10.3) / 1.2 = -1.08, and for the upper limit (10 cm) is (10 - 10.3) / 1.2 = -0.25. For Company B, the z-score for the lower limit is (9 - 10.6) / 0.9 = -1.78, and for the upper limit is (10 - 10.6) / 0.9 = -0.67.

Next, we use a standard normal distribution table (or a calculator) to find the proportion of the area under the curve between these z-scores. The proportion for Company A is P(-1.08 ≤ Z ≤ -0.25) = 0.3093 - 0.3997 = 0.0906, and for Company B is P(-1.78 ≤ Z ≤ -0.67) = 0.0375 - 0.2525 = 0.2150.

Therefore, Company B provides a higher proportion of usable onions compared to Company A.

Answer 2

Follows are the solution to this question:

Step-by-step explanation:

Let assume that X and Y reflect the diameters of onions by companies A and B.

`\to P(9<X<10) = P((9-10.3)/(1.2) < Z < (10-10.3)/(1.2) )\\\\`

`= P(-1.0833 < Z < -0.25)\\\\= P(Z < -0.25) - P(Z<-1.0833)\\\\= 0.2620\\\\\\`

`\to P(9<Y<10) = P((9-10.6)/(0.9) < Z < (10-10.6)/(0.9) )\\\\`

`= P(-1.7778 < Z < -0.6667)\\\\= 0.2148\\\\`

As a result, onions through company A are chosen and they're more likely to be accepted.