A production process produces an item, On average, 13% of all items produced are defective. Each item is inspected before being shipped, and the inspector misclassifies an item 12% of the time. What proportion of the items will be "classified as good"?
I don't even need an answer, just an explanation would be great
out of a thousand items:
130 are broken , 870 are ok
out of the 130 broken, 15.6 are mistakenly labeled good
out of the 870 good, 20.4 are mistakenly labeled broken, so 849.6 classified good
so
15.6 + 849.6 = 865.2 are labeled good
100 * (865.2/1000) = 86.5 % are classified "good"
(although in fact 15.6 or 100(15.6/849.6)= 1.8% of those are actually broken and mistakenly labeled good)
Oh, so you want an explanation? Well, let me put my clown glasses on and try to explain this in a funny way!
Imagine you're at a circus, and there's a clown making balloon animals. Let's say this clown is not very skilled, so sometimes the balloons come out defective. Oh no!
Now, imagine there's an inspector at the end of the line who checks all the balloon animals before they are given to the audience. But here's the catch – this inspector is a bit of a silly clown himself, and sometimes he misjudges whether a balloon is good or defective. Oops, wrong call!
So, we have two sources of potential errors here – the original defective balloons and the inspector's misjudgments. The question is asking about the proportion of items (balloons) that will be classified as good.
To find this out, we need to consider both the original defect rate and the misclassification rate. If the original defect rate is 13% and the inspector misclassifies 12% of the time, then it means that not all defective balloons will be correctly identified as defective. Some will slip through and be classified as good.
Mathematically, we can think of it as a two-step process. First, we take the 13% defective balloons and multiply it by the 88% chance that the inspector correctly identifies them as defective. This gives us the proportion of correctly identified defective balloons.
Then, we take the remaining 87% of balloons that were not defective in the first place and multiply it by the 12% chance that the inspector misclassifies them as defective. This gives us the proportion of good balloons that are mistakenly classified as defective.
Finally, we add these two proportions together to get the total proportion of items that will be classified as good. Phew!
So, to summarize, the proportion of items that will be "classified as good" is a combination of both the original defect rate and the inspector's misclassification rate. It's like a circus of errors, but hopefully, this explanation brings a smile to your face!
To determine the proportion of items that will be classified as "good," we need to consider two factors: the proportion of items that are actually good and the misclassification rate of the inspector.
First, let's consider the proportion of items that are actually good. Given that 13% of all items produced are defective, the proportion of good items would be 1 - 0.13 = 0.87, or 87%.
Next, we need to take into account the misclassification rate of the inspector, which is 12%. This means that the inspector misclassifies an item as good 12% of the time. So, if 87% of the items are actually good, 12% of the 87% would be misclassified as good.
To find the proportion of items that will be classified as good, we need to subtract the proportion of items that will be misclassified as good from the proportion of items that are actually good.
Proportion of items classified as good = Proportion of items actually good - Proportion of items misclassified as good.
Proportion of items classified as good = 0.87 - (0.12 * 0.87)
Proportion of items classified as good = 0.87 - 0.1044
Proportion of items classified as good ≈ 0.7656, or approximately 76.56%.
Therefore, approximately 76.56% of the items will be classified as "good."
To find the proportion of items that will be "classified as good," we need to consider two scenarios: when an item is actually good and when an item is defective.
Let's start by considering the scenario when an item is actually good. The probability of an item being good is given by 1 minus the proportion of defective items. In this case, the proportion of good items would be 1 - 0.13 (since 13% are defective).
Now let's consider the scenario when an item is defective. In this case, the inspector might misclassify the item as good. The probability of misclassification is given as 12%. Hence, the proportion of defective items that will be classified as good would be 0.13 multiplied by 0.12.
To find the overall proportion of items that will be "classified as good," we need to add the proportions of good items and the incorrectly classified defective items. Mathematically, it can be represented as:
Proportion of "classified as good" items = Proportion of good items + Proportion of incorrectly classified defective items
P("classified as good") = (1 - 0.13) + (0.13 * 0.12)
Therefore, by adding the two values, we can determine the proportion of items that will be "classified as good."