Question

The amount of time a certain brand of light bulb lasts is normally distributed with a mean of 1900 hours and a standard deviation of 20 hours. Using the empirical rule, what percentage of light bulbs last between 1840 hours and 1960 hours? a) 68% b) 95% c) 99.7% d) 99%

Answer 1

Approximately 95.44% of the light bulbs will last between 1840 hours and 1960 hours.

Step-by-step explanation:

To find the percentage of light bulbs that last between 1840 hours and 1960 hours, we can use the empirical rule, also known as the 68-95-99.7 rule. According to this rule, approximately 68% of the light bulbs will fall within one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will fall within three standard deviations.

Calculating the Standard Deviation

We are given that the mean is 1900 hours and the standard deviation is 20 hours. One standard deviation below the mean is 1900 - 20 = 1880 hours, and one standard deviation above the mean is 1900 + 20 = 1920 hours.

Calculating the Percentage

To find the percentage of light bulbs that fall between 1840 hours and 1960 hours (two standard deviations below and above the mean), we can subtract the percentage of light bulbs that fall below 1880 hours from the percentage that fall below 1960 hours.

Percentage = P(X < 1960) - P(X < 1880)

Using the standard normal distribution table or a calculator, we can find these probabilities:

P(X < 1960) ≈ 0.9772

P(X < 1880) ≈ 0.0228

Therefore, the percentage of light bulbs that last between 1840 hours and 1960 hours is approximately 0.9772 - 0.0228 = 0.9544, or 95.44%.