Company A packaged bolts in boxes that are normally distributed with a mean of 312 bolts and a standard deviation of 4.1 bolts. Company B packages bolts in boxes that are normally distributed with a mean of 290 bolts and a standard deviation of 3.7 bolts. Which Company is more likely to produce a box of exactly 300 bolts? Explain your answer using Z scores.
To determine which company is more likely to produce a box of exactly 300 bolts, we can use the concept of Z scores. A Z score measures how many standard deviations a data point is away from the mean.
For Company A:
Mean (μ) = 312 bolts
Standard deviation (σ) = 4.1 bolts
To calculate the Z score for a box of exactly 300 bolts in Company A, we use the formula:
Z = (X - μ) / σ
Substituting the values, we get:
Z = (300 - 312) / 4.1
Calculating this, we find that the Z score for Company A is approximately -2.93.
For Company B:
Mean (μ) = 290 bolts
Standard deviation (σ) = 3.7 bolts
To calculate the Z score for a box of exactly 300 bolts in Company B, we use the formula:
Z = (X - μ) / σ
Substituting the values, we get:
Z = (300 - 290) / 3.7
Calculating this, we find that the Z score for Company B is approximately 2.70.
Now, let's interpret the Z scores:
A negative Z score indicates that the value is below the mean, while a positive Z score indicates that the value is above the mean. The magnitude of the Z score represents how far away the value is from the mean in terms of standard deviations.
In this case, Company A has a Z score of -2.93, and Company B has a Z score of 2.70. Since both Z scores are quite large (in absolute terms), it suggests that both companies' chances of producing a box of exactly 300 bolts are relatively low, as the value is significantly different from their respective means.
However, based on the Z scores, we can conclude that Company B is more likely to produce a box of exactly 300 bolts. This is because Company B's Z score (2.70) is closer to zero, which indicates a higher probability of observing a value around the mean. On the other hand, Company A's Z score (-2.93) indicates a lower probability of observing a value close to the mean.
Therefore, Company B is more likely to produce a box of exactly 300 bolts.