To find the probability that a bag of sugar selected at random is rejected, we need to calculate the area under the normal distribution curve to the left of the rejection mass (995 grams). This is essentially calculating the cumulative probability for a mass less than 995 grams.
Step 1: Normalize the variable using the Z-score.
The Z-score transforms the random variable (the mass of the bags) into a standard normal distribution with a mean of 0 and a standard deviation of 1. The formula for finding the Z-score of an observation is:
\[ z = \frac{(X - \mu)}{\sigma} \]
Where:
- \( X \) is the value of the observation, in our case, the rejection mass of 995 grams.
- \( \mu \) is the mean of the distribution, which is given as 1000 grams.
- \( \sigma \) is the standard deviation of the distribution, provided as 3.5 grams.
Step 2: Calculate the Z-score.
\[ z_{rejection} = \frac{(995 - 1000)}{3.5} \]
\[ z_{rejection} = \frac{-5}{3.5} \]
\[ z_{rejection} = -1.4286 \]
(We keep several decimal places to maintain precision for our calculations.)
Step 3: Use standard normal distribution tables or a calculator to find the cumulative probability for the Z-score.
Now that we have the Z-score representing the rejection mass, we look up or calculate the cumulative probability for this Z-score in a standard normal distribution.
The cumulative probability for a Z-score of -1.4286 indicates the probability that a randomly selected bag of sugar weighs less than 995 grams.
Using a standard normal distribution table or calculator, the cumulative probability associated with a Z-score of -1.4286 might not be directly listed, so you would find the closest value in the table. For example, if you have a Z-table that lists probabilities to two decimal places, you might find Z = -1.43. Alternatively, you might need to use linear interpolation between two values or use a calculator that can determine the exact cumulative probability for the Z-score.
Typically, the cumulative probability associated with a Z-score of -1.43 is approximately 0.0764.
Therefore, the probability that a bag selected at random is rejected (i.e., its mass is less than 995 grams) is about 0.0764, or 7.64%.