2) The lifetimes of light bulbs of a particular type are normally distributed with a mean of 370 hours and a standard deviation of 5 hours. What percentage of bulbs has lifetimes that lie within 1 standard deviation of the mean on either side?
check the 68% rule
To find the percentage of bulbs with lifetimes that lie within 1 standard deviation of the mean on either side, we can use the properties of a normal distribution.
First, let's calculate the range within 1 standard deviation of the mean:
Lower Bound = Mean - 1 * Standard Deviation
Upper Bound = Mean + 1 * Standard Deviation
Lower Bound = 370 - 1 * 5 = 365
Upper Bound = 370 + 1 * 5 = 375
Next, we need to find the percentage of bulbs with lifetimes between the lower and upper bound. To do this, we will calculate the area under the normal distribution curve between these two values.
The area under the curve represents the percentage of bulbs within that range. To find this area, we need to calculate the z-scores for the lower and upper bounds.
Z-score = (X - Mean) / Standard Deviation
For the lower bound:
Z-score = (365 - 370) / 5 = -1
For the upper bound:
Z-score = (375 - 370) / 5 = 1
Now, we can use a standard normal distribution table or a statistical software to look up the corresponding cumulative probability associated with these z-scores.
From the standard normal distribution table, we find that the cumulative probability for a z-score of -1 is 0.1587, and the cumulative probability for a z-score of 1 is 0.8413.
To find the percentage of bulbs within the range, we subtract the lower cumulative probability from the upper cumulative probability:
Percentage = (0.8413 - 0.1587) * 100 = 68.26 %
Therefore, approximately 68.26% of bulbs will have lifetimes that lie within 1 standard deviation of the mean on either side.