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:(give 4 decimal places for each answer)
a. Less than 250 hours
b. Between 225 and 375 hours
c. More than 400 hours
I know the formula. X-mean/SD. I need help checking my answers!
this site should help:
http://davidmlane.com/hyperstat/z_table.html
To solve these problems using the given formula, we need to convert each value to a z-score. The z-score formula is (X - mean) / SD.
a. Less than 250 hours:
Z = (250 - 300) / 75 = -0.6667
Using a standard normal distribution table or calculator, we find that the probability of a Z-score less than -0.6667 is 0.2525. So, the probability that the battery will last less than 250 hours is approximately 0.2525.
b. Between 225 and 375 hours:
For the lower bound:
Z_lower = (225 - 300) / 75 = -1.0000
For the upper bound:
Z_upper = (375 - 300) / 75 = +1.0000
Using the standard normal distribution table or calculator, we find that the probability of a Z-score less than -1.0000 is 0.1587, and the probability of a Z-score less than 1.0000 is 0.8413.
To find the probability between two values, we subtract the lower probability from the upper probability:
P(225 < X < 375) = P(X < 375) - P(X < 225) = 0.8413 - 0.1587 = 0.6826. So, the probability that the battery will last between 225 and 375 hours is approximately 0.6826.
c. More than 400 hours:
Z = (400 - 300) / 75 = +1.3333
Using the standard normal distribution table or calculator, we find that the probability of a Z-score more than 1.3333 is 0.0918. So, the probability that the battery will last more than 400 hours is approximately 0.0918.
These are the step-by-step calculations for each problem. Let me know if you need any further assistance or want to verify your answers!
To solve these problems, you will need to use the concept of z-scores, which is calculated as (X - μ) / σ. Here, X represents the value you want to find the probability for, μ represents the mean, and σ represents the standard deviation.
a. To find the probability that the battery will last less than 250 hours, you need to find the area under the normal curve to the left of 250 hours. Calculate the z-score using the formula:
z = (250 - 300) / 75
Now, you need to find the probability corresponding to this z-score. You can use a z-table or a statistical calculator to find the cumulative probability. For example, using a z-table, find the value closest to -1.33 (the z-score calculated), you would find that the corresponding area is 0.0918.
So, the probability that the battery will last less than 250 hours is approximately 0.0918 (rounded to 4 decimal places).
b. To find the probability that the battery will last between 225 and 375 hours, you need to find the area under the normal curve between those two values.
First, calculate the z-scores for both values:
z1 = (225 - 300) / 75
z2 = (375 - 300) / 75
Using the z-table, find the probabilities corresponding to z1 and z2. For example, you would find that the area to the left of z1 is 0.1056 and the area to the left of z2 is 0.8944.
The probability between these two values is the difference between the two probabilities:
P(225 < X < 375) = 0.8944 - 0.1056 = 0.7888
So, the probability that the battery will last between 225 and 375 hours is approximately 0.7888 (rounded to 4 decimal places).
c. To find the probability that the battery will last more than 400 hours, you need to find the area under the normal curve to the right of 400 hours.
Calculate the z-score:
z = (400 - 300) / 75
Using the z-table, find the probability corresponding to the z-score. For example, you would find that the area to the left of z is 0.9332.
The probability that the battery will last more than 400 hours is 1 minus this value:
P(X > 400) = 1 - 0.9332 = 0.0668
So, the probability that the battery will last more than 400 hours is approximately 0.0668 (rounded to 4 decimal places).
Note: Make sure to use the correct tail (left or right) of the normal curve depending on the question.