The Lifetimes of a certain brand of photographic light are normally distributed with the mean of 210 h and a standard deviation of 50 h.
What percent of lights will need to be replaced within 233 h?
Out of 2000 lights, how many will have a lifetime between 200 h and 400 h?
To answer these questions, we can use the concept of the standard normal distribution. We will need to calculate the z-scores and use the z-table (also known as the standard normal table) to determine the probabilities.
Question 1: What percent of lights will need to be replaced within 233 h?
To find the percentage of lights that will need to be replaced within 233 hours, we need to find the area under the normal distribution curve to the left of 233 hours. First, we calculate the z-score using the formula:
z = (x - μ) / σ
Where:
- x is the value (233 hours in this case)
- μ is the mean (210 hours in this case)
- σ is the standard deviation (50 hours in this case)
Substituting the given values, we have:
z = (233 - 210) / 50
z = 23 / 50
z = 0.46
Next, we use the z-score to find the corresponding area in the standard normal table. The standard normal table gives us the area to the left of a given z-score. Looking up the z-score of 0.46 in the table, we find that the area to the left is approximately 0.6772.
This means that approximately 67.72% of lights will need to be replaced within 233 hours.
Question 2: Out of 2000 lights, how many will have a lifetime between 200 h and 400 h?
To determine the number of lights that will have a lifetime between 200 and 400 hours, we need to find the area under the normal distribution curve between these two values. We calculate the z-scores for 200 and 400 hours using the same formula as before:
z1 = (200 - 210) / 50 = -0.20
z2 = (400 - 210) / 50 = 3.80
Next, we use the z-scores to find the corresponding areas in the standard normal table. We find the area to the left of z1 and z2 and subtract one from the other to get the area between them. Using the standard normal table, we find that the area to the left of -0.20 is approximately 0.4207 and the area to the left of 3.80 is approximately 0.9997. Therefore, the area between z1 and z2 is: 0.9997 - 0.4207 = 0.5790.
To find the number of lights that fall within this range, we multiply the area (0.5790) by the total number of lights (2000):
Number of lights = 0.5790 * 2000 = 1158
Therefore, approximately 1158 lights will have a lifetime between 200 hours and 400 hours.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.