A company makes batteries with an average life span of 300 hours with a standard deviation of 75 hours. Assuming the distribution is approximated by a normal curve fine the probability that the battery will last .
Less than 250 hours
b. Between 225 and 375 hours
c. More than 400 hours
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 related to the Z scores.
To find the probabilities for each scenario, we need to use z-scores and the standard normal distribution.
a. To find the probability that the battery will last less than 250 hours:
1. Calculate the z-score for 250 hours: z = (250 - 300) / 75 = -0.67
2. Look up the z-score in the z-table or use a calculator to find the corresponding cumulative probability: P(Z < -0.67) = 0.2514
Therefore, the probability that the battery will last less than 250 hours is approximately 0.2514.
b. To find the probability that the battery will last between 225 and 375 hours:
1. Calculate the z-scores for 225 and 375 hours: z1 = (225 - 300) / 75 = -1.00, z2 = (375 - 300) / 75 = 1.00
2. Find the cumulative probability for each z-score: P(Z < -1.00) = 0.1587, P(Z < 1.00) = 0.8413
3. Subtract the smaller probability from the larger probability to find the probability between the two z-scores: P(-1.00 < Z < 1.00) = P(Z < 1.00) - P(Z < -1.00) = 0.8413 - 0.1587 = 0.6826
Therefore, the probability that the battery will last between 225 and 375 hours is approximately 0.6826.
c. To find the probability that the battery will last more than 400 hours:
1. Calculate the z-score for 400 hours: z = (400 - 300) / 75 = 1.33
2. Find the cumulative probability for the z-score: P(Z > 1.33) = 1 - P(Z < 1.33) (using a calculator or z-table)
= 1 - 0.908 = 0.092
Therefore, the probability that the battery will last more than 400 hours is approximately 0.092.