Use the table to answer the question. Note: Round z-scores to the nearest hundredth and then find the required A values using the table.
A manufacturer of light bulbs finds that one light bulb model has a mean life span of 1025 h with a standard deviation of 88 h. What percent of these light bulbs will last as follows? (Round your answers to one decimal place.)
(a) at least 940 h
%
(b) between 850 and 900 h
%
To find the percentages, we need to use z-scores and the standard normal distribution table.
A z-score measures the number of standard deviations a value is from the mean.
(a) To find the percentage of light bulbs that will last at least 940 hours, we need to calculate the z-score for 940 hours.
The formula for calculating the z-score is: z = (x - μ) / σ
Where:
x = the value (940 hours)
μ = the mean (1025 hours)
σ = the standard deviation (88 hours)
Substituting the given values into the formula, we have:
z = (940 - 1025) / 88
Calculate this z-score using a calculator or manually:
z ≈ -0.97
Next, we look up the z-score in the standard normal distribution table. The table provides the area under the curve to the left of the z-score. However, we want the area to the right of the z-score since we want the percentage of light bulbs that will last at least 940 hours.
To find the area to the right of the z-score, subtract the area to the left of the z-score from 1:
Area to the right = 1 - Area to the left
Using the table, we find that the area to the left of -0.97 is 0.1664.
Therefore, the area to the right of -0.97 is approximately 1 - 0.1664 = 0.8336.
Lastly, multiply this area by 100 to convert it to a percentage:
% = 0.8336 * 100
The percentage of light bulbs that will last at least 940 hours is approximately 83.4%.
(b) To find the percentage of light bulbs that will last between 850 and 900 hours, we need to calculate the z-scores for these two values separately and find the area in between.
For 850 hours:
z = (850 - 1025) / 88
z ≈ -1.99
For 900 hours:
z = (900 - 1025) / 88
z ≈ -1.43
Using the standard normal distribution table, we find that the area to the left of -1.99 is about 0.0238, and the area to the left of -1.43 is about 0.0764.
To find the area between these two z-scores, subtract the smaller area from the larger area:
Area between = Area to the right of the smaller z-score - Area to the right of the larger z-score
Using the table, we find that the area to the right of -1.99 is approximately 1 - 0.0238 = 0.9762, and the area to the right of -1.43 is approximately 1 - 0.0764 = 0.9236.
Therefore, the area between these two z-scores is approximately 0.9762 - 0.9236 = 0.0526.
Lastly, multiply this area by 100 to convert it to a percentage:
% = 0.0526 * 100
The percentage of light bulbs that will last between 850 and 900 hours is approximately 5.3%.