Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of and a standard deviation of 0.04 ounce. Find the probability that a randomly selected bottle contains between 12.31 and 12.37 ounces.
You have not said what the mean value is is. A number was left out.
The mean is 12.41 ounces
All information included
To find the probability that a randomly selected bottle contains between 12.31 and 12.37 ounces, we can use the properties of the normal distribution.
1. Standardize the values: First, we need to standardize the given values 12.31 and 12.37. We can do this by using the formula:
z = (x - μ) / σ,
where z represents the standard score, x is the given value, μ is the mean, and σ is the standard deviation.
So, for 12.31:
z1 = (12.31 - μ) / σ,
and for 12.37:
z2 = (12.37 - μ) / σ.
2. Look up the z-scores: Once we have the z-scores for the given values, we need to determine the corresponding probabilities using a standard normal distribution table or a calculator.
The table or calculator will provide the probability associated with each z-score.
3. Calculate the final probability: Finally, we calculate the final probability by finding the difference between the two probabilities from step 2. Since the values are between 12.31 and 12.37, we need to find the area under the normal curve between these two z-scores.
P(12.31 ≤ X ≤ 12.37) = P(z1 ≤ Z ≤ z2),
where Z represents the standard normal random variable.
By following these steps, you should be able to find the probability that a randomly selected bottle contains between 12.31 and 12.37 ounces using the given mean and standard deviation.