To find the probability of randomly selecting a cigarette with 0.572 g of nicotine or less, we can use the Z-score formula. The Z-score is calculated using the following formula:
\[ Z = \frac{(X - \mu)}{\sigma} \]
where:
- \( X \) is the value we are interested in (0.572 g),
- \( \mu \) is the mean (0.963 g), and
- \( \sigma \) is the standard deviation (0.301 g).
Substituting the given values:
\[ Z = \frac{(0.572 - 0.963)}{0.301} = \frac{-0.391}{0.301} \approx -1.296 \]
Next, we need to find the probability corresponding to the Z-score of -1.296. We can look this value up in the standard normal distribution table or use a calculator or software that provides cumulative probabilities for the normal distribution.
Using standard normal distribution tables or a calculator:
- The cumulative probability \( P(Z < -1.296) \) is approximately 0.0974.
Thus, the probability of randomly selecting a cigarette with 0.572 g of nicotine or less is:
\[ P(X < 0.572) \approx 0.0974 \]
Finally, rounding to four decimal places, we get:
\[ P(X < 0.572 g) = 0.0974 \]