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?
Z = ± 1
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the Z scores. Multiply by 100.
To find the percentage of bulbs that have lifetimes within 1 standard deviation of the mean on either side, we can use the empirical rule for normally distributed data.
According to the empirical rule, approximately 68% of the data falls within 1 standard deviation of the mean in a normal distribution.
Therefore, the percentage of bulbs with lifetimes within 1 standard deviation of the mean on either side is approximately 68%.
To find the percentage of bulbs that have lifetimes within 1 standard deviation of the mean, we can use the properties of the normal distribution.
Step 1: Calculate the range of lifetimes within 1 standard deviation of the mean.
Since we know that the mean is 370 hours and the standard deviation is 5 hours, we can calculate the range as follows:
Lower Range = Mean - Standard Deviation
Lower Range = 370 - 5 = 365 hours
Upper Range = Mean + Standard Deviation
Upper Range = 370 + 5 = 375 hours
Step 2: Calculate the percentage of bulbs within this range.
To find the percentage of bulbs within the range, we need to calculate the area under the normal distribution curve between the lower and upper range.
Since the distribution is symmetric, we'll calculate the area to the right of the lower range and double it to include both sides.
To find this area, we can use a Z-table or a statistical software. Let's use a Z-table.
Step 3: Convert the range to Z-scores.
To use the Z-table, we need to convert the range into Z-scores. We can use the formula:
Z = (X - Mean) / Standard Deviation
For the lower range:
Z_lower = (365 - 370) / 5 = -1
For the upper range:
Z_upper = (375 - 370) / 5 = 1
Step 4: Look up the Z-scores in the Z-table.
Using the Z-table, we can find the area to the right of -1 (Z_lower) and to the right of 1 (Z_upper). Let's assume we find that the area to the right of -1 is 0.8413 and the area to the right of 1 is 0.1587.
Step 5: Calculate the percentage.
Since the distribution is symmetric, the total percentage of bulbs within 1 standard deviation of the mean is double the area we found in Step 4.
Total Percentage = (0.8413 + 0.1587) * 2 = 0.998 * 2 = 0.996 (approximately)
Therefore, approximately 99.6% of bulbs have lifetimes within 1 standard deviation of the mean.