Question

for a brand, the life of light bulbs (b) are normally distributed with a mean of 750 hours and a standard deviation of 75 hours. find the probability that a given light bulb lasts between 675 and 900 hours. p(675 < b < 900) = ?% be sure to use the 68% - 95% - 99.7% rule and do not round

Explanation:

Step1: Analyze the position of 675

The mean is \( \mu = 750 \) and the standard deviation is \( \sigma=75 \). For \( x = 675 \), \( \mu-\sigma=750 - 75=675 \).

Step2: Analyze the position of 900

For \( x = 900 \), \( \mu + 2\sigma=750+2\times75 = 750 + 150=900 \).

Step3: Calculate the probability

The \( 68 - 95 - 99.7\) rule states that:

• The probability within \( \mu\pm\sigma \) is \( 68\% \) (i.e., \( P(675\lt b\lt825) = 68\% \))
• The probability within \( \mu\pm2\sigma \) is \( 95\% \), so the probability within \( 825\lt b\lt900 \) is \( \frac{95\% - 68\%}{2}=13.5\% \)

Then \( P(675\lt b\lt900)=P(675\lt b\lt825)+P(825\lt b\lt900) \)

\( P(675\lt b\lt900)=68\%+13.5\% \)

Answer:

\( 81.5\% \)