We're dealing with refrigerators and we're told that that's approximately normal. And we know that the average life before replacement is supposed to be 14 years. And we know that the Part that is including 95 is from nine years to 19 years. So that's going to mean this is equivalent to two standard deviations and that difference is five years. So two standard deviations is equal to nine, I'm sorry, it's equal to five, which means the standard deviation is equivalent to because this distance is five from here to here or from here to here.
And so we know a standard deviation would be considered 2.5 years, so that's what we will use, will use this mean and the standard deviation. So we want to answer that's for part a part B, we want to find what's the likelihood that someone has to replace it Before 11 years. So how many standard deviations, is that a way you can see 11 years, it's going to end up being approximately right in here and so it's a negative Z value and we have that 11 minus the mean divided by the standard deviation. Well round that off to two decimal places, so that becomes negative three divided by 2.5 And negative three divided by 2.5 groups, I have three divided by 2.5, I don't think there's only 1.2, So Z is less than negative 1.2 and look up negative 1.2. And we find that that probability is .1151.
Now part C, we want to know what's the likelihood that it will be replaced. And and let me just verify and less than I believe it said for part C, Let's look here that it will uh they'll keep it more than 18 years, So more than 18 years and we need to convert that to a Z value. So we replace the 18 right there and 18, which is 18 minus the mean is four And then divided by 2.5. That gives us is the value of 1.6. So I'm not going to look up 1.6, I'm gonna look up negative 1.6.
So that area below negative 1.6 will be that same answer. So the likelihood that you're about right here up in this tale is a little over 5%. Now, part D They say that they want to give a guarantee for a length of time for just the lower 5%. So they definitely don't want to end up saying that the guarantee is for 14 years because then they'll have to replace happy though. So we want to find out what that cut off point that has .05 here.
So we need to look up in our table and do the inverse either whether calculated, do the inverse normal. And this is a kind of a common number to come up. We'll find in statistics and we won't find this one will find one very close to it. And this one is negative 1.64, and this one is negative 1.6 five. And so it's very common for us just to use that halfway in between.
And so this is our Z score And we know our mean is 14 and our standard deviation we're using is 2.5. So if we take 14 -1.645 Times 2.5, that's going to give us that length. Yeah, let me quick type that in. And when I do, I get 9.8897. And I don't know that.
The question says I wouldn't say roughly 10 months or 9.9. I'm sorry. It's in years. So If they guarantee it for like less and if it lasts less than 10 years, they're going to have to replace about 5%. Mhm.