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, we will use the properties of the normal distribution. The mean \(\mu\) is 1400 hours and the standard deviation \(\sigma\) is 80 hours.

We will first standardize the values of 1200 hours and 1510 hours into z-scores using the formula:

\[
z = \frac{(X - \mu)}{\sigma}
\]

**Step 1: Calculate the z-score for 1200 hours.**

\[
z_{1200} = \frac{(1200 - 1400)}{80} = \frac{-200}{80} = -2.5
\]

**Step 2: Calculate the z-score for 1510 hours.**

\[
z_{1510} = \frac{(1510 - 1400)}{80} = \frac{110}{80} = 1.375
\]

**Step 3: Use the z-scores to find the corresponding probabilities from the standard normal distribution table.**

- For \(z = -2.5\):
- The cumulative probability \(P(Z < -2.5)\) is approximately \(0.0062\).

- For \(z = 1.375\):
- The cumulative probability \(P(Z < 1.375)\) is approximately \(0.9157\).

**Step 4: Find the probability of the light bulbs lasting between 1200 and 1510 hours.**

The percentage of light bulbs that last between 1200 hours and 1510 hours is found by calculating:

\[
P(1200 < X < 1510) = P(Z < 1.375) - P(Z < -2.5)
\]

Substituting our calculated probabilities:

\[
P(1200 < X < 1510) \approx 0.9157 - 0.0062 = 0.9095
\]

**Step 5: Convert to percentage and round to the nearest tenth.**

\[
0.9095 \times 100 \approx 90.95\%
\]

Rounded to the nearest tenth, the percentage of light bulbs that last between 1200 hours and 1510 hours is:

\[
\boxed{91.0\%}
\]