Question

The amount of time a certain brand of light bulb lasts is normally distributed with a mean of 1400 hours and a standard deviation of 80 hours. what percentage light bulbs last between 1200 hours and 1510 hours, to the nearest tenth?

Answer 1

To find the percentage of light bulbs that last between 1200 hours and 1510 hours, calculate the z-scores for the given values. Then, use the z-scores to find the corresponding probabilities from the z-table. Finally, subtract the probability for 1200 hours from the probability for 1510 hours to find the percentage of bulbs that last between the two values.

Step-by-step explanation:

To find the percentage of bulbs that last between 1200 hours and 1510 hours, we need to find the area under the normal distribution curve between these two values.

First, we calculate the z-scores for the given values using the formula:

z = (x - mean) / standard deviation

For 1200 hours: z = (1200 - 1400) / 80 = -0.25

For 1510 hours: z = (1510 - 1400) / 80 = 1.375

Next, we look up these z-scores in the z-table to find the corresponding probabilities. From the table, we find that the probability for a z-score of -0.25 is 0.4013 and the probability for a z-score of 1.375 is 0.9154.

Finally, we subtract the probability for 1200 hours from the probability for 1510 hours to find the percentage of bulbs that last between the two values: 0.9154 - 0.4013 = 0.5141. Converting to a percentage, we round to the nearest tenth: 51.4%.

Answer 2

Approximately 90.92% of the light bulbs last between 1200 and 1510 hours.

To find the percentage of light bulbs that last between 1200 and 1510 hours, we need to calculate the z-scores for each value and then use the standard normal distribution table.

First, let's calculate the z-score for 1200 hours:

z = (1200 - 1400) / 80

z = -200 / 80

z = -2.5

Next, let's calculate the z-score for 1510 hours:

z = (1510 - 1400) / 80

z = 110 / 80

z = 1.375

Now, we can use the standard normal distribution table to find the area under the curve between these two z-scores.

From the table, the area to the left of -2.5 is approximately 0.0062, and the area to the left of 1.375 is approximately 0.9154.

To find the percentage between these two z-scores, we subtract the smaller area from the larger area:

0.9154 - 0.0062 = 0.9092

Remember to round your final answer to the nearest tenth, which gives us 90.9%.