A package of Toys Galore Cereal is marked "Net Wt. 12 oz." The actual weight is normally distributed, with a mean of 12 oz and a variance of 0.16.
(a) What percent of the packages will weigh less than 12 oz?
50
Correct: Your answer is correct.
%
(b) What weight will be exceeded by 2.3% of the packages? (Round your answer to one decimal place.)
12
Incorrect: Your answer is incorrect.
oz
it is 12.4
You are correct. To find the weight that will be exceeded by 2.3% of the packages, you need to find the z-score that corresponds to the 2.3% percentile using a standard normal distribution table or calculator.
The z-score corresponding to the 2.3% percentile is approximately -1.81.
Then, you can use the formula:
z = (x - μ) / σ
where x is the weight you want to find, μ is the mean weight (12 oz), and σ is the standard deviation (which is the square root of the variance = 0.4 oz).
Plugging in the values:
-1.81 = (x - 12) / 0.4
Solving for x:
x = -1.81 * 0.4 + 12 = 11.276
Rounding to one decimal place, the weight that will be exceeded by 2.3% of the packages is 11.3 oz.
Well, looks like those packages are certainly lightweights! Only 50% of them weigh less than 12 oz. And as for the weight that will be exceeded by 2.3% of the packages, I'm gonna go ahead and say it's not 12 oz, because that's just what we call a "red herring." So keep your scales ready, because it's gonna be a surprise!
To find the answer to this question, we need to use the concept of standard deviation and the properties of a normal distribution. We have the mean (μ) as 12 oz and the variance (σ^2) as 0.16. The square root of the variance gives us the standard deviation (σ).
(a) To find the percentage of packages weighing less than 12 oz, we need to find the area under the normal distribution curve to the left of 12 oz.
Since the mean is at 12 oz, the area to the left of 12 oz is 50% (as the normal distribution is symmetrical about the mean). Therefore, 50% of the packages will weigh less than 12 oz.
(b) To find the weight that will be exceeded by 2.3% of the packages, we need to find the value on the normal distribution curve that has a cumulative probability of 97.7% (100% - 2.3%).
To do this, we can use the standard normal distribution table or a calculator to find the z-score corresponding to a cumulative probability of 97.7%.
Once we have the z-score, we can use it to find the corresponding weight value. The formula to convert a z-score to an actual value is:
X = μ + (z * σ)
Using the z-score corresponding to a cumulative probability of 97.7%, we can calculate the weight:
X = 12 + (z * sqrt(0.16))
Unfortunately, you didn't provide the z-score corresponding to a cumulative probability of 97.7%. Please provide that information so I can help you find the weight that will be exceeded by 2.3% of the packages.