Final answer:
The probability of a color television picture tube lasting (a) more than 10 years is approximately 9.85%, (b) less than 5 years is approximately 1.11%, and (c) between 5 and 10 years is approximately 8.74%.
Step-by-step explanation:
To determine the probability for different lifetimes of color television picture tubes, we use the normal distribution. The mean lifetime is 8.2 years and the standard deviation is 1.4 years.
a) To find the percentage of tubes that last more than 10 years, we need to calculate the area under the normal curve to the right of 10. We standardize the value 10 by subtracting the mean from it and dividing by the standard deviation.
This gives us a standardized value of (10 - 8.2) / 1.4 = 1.29. Using a Z-table or calculator, we can find the area to the right of 1.29, which is approximately 0.0985. Therefore, about 9.85% of the tubes last more than 10 years.
b) To find the percentage of tubes that last less than 5 years, we standardize the value 5 and find the area to the left of it. The standardized value is (5 - 8.2) / 1.4 = -2.29. Looking up the area to the left of -2.29, we get approximately 0.0111. So, about 1.11% of the tubes last less than 5 years.
c) To find the percentage of tubes that last between 5 and 10 years, we subtract the area to the left of 5 (0.0111) from the area to the left of 10 (0.0985). This gives us approximately 0.0874, or 8.74% of the tubes.