Question

A type of battery-operated led lights has a known mean lifetime 7.9 hours with standard deviation 0.5. Assuming that the lifetimes of these led lights have a nearly symmetric/bell-curve distribution, find the percent (%) of these led lights having lifetime between 7.9 and 8.9 hours.

Answer 1

Approximately 68% of the LED lights have a lifetime between 7.4 and 8.4 hours.

Step-by-step explanation:

To find the percent of LED lights having a lifetime between 7.9 and 8.9 hours, we need to calculate the area under the bell curve within that range. Since the distribution is nearly symmetric, we can use the Empirical Rule to estimate the probability.

According to the Empirical Rule, approximately 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

In this case, the lifetime range of 7.9 to 8.9 hours is within one standard deviation of the mean (7.9 ± 0.5).

• The mean value plus one standard deviation is 7.9 + 0.5 = 8.4 hours
• The mean value minus one standard deviation is 7.9 - 0.5 = 7.4 hours

Therefore, approximately 68% of the LED lights have a lifetime between 7.4 and 8.4 hours.

Answer 2

47.5% of these led lights have lifetime between 7.9 and 8.9 hours.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 7.9 hours

Standard deviation = 0.5 hours

The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.

Lifetime between 7.9 and 8.9 hours:

7.9 hours is the mean.

8.9 = 7.9 + 2*0.5

So 8.9 hours is two standard deviations above the mean.

Of the 50% of the measures that are above the mean, 95% are between the mean of 7.9 and two standard deviations above the mean(8.9). So

0.5*0.95 = 0.475

47.5% of these led lights have lifetime between 7.9 and 8.9 hours.