Area under a normal curve: How long should the guarantee be?

[ChrisBlack]

Homework Statement

The mean is 17.8 and the standard deviation is 5.2. The shop does not want to give more than 1% of its customers free service, how long should the guarantee be? (cut the problem short, the rest was just story)

Homework Equations

I think the equation I would use is X = Z(standard deviation) + Mean

The Attempt at a Solution

I changed 1% into the Z score and got -2.33 and plugged that into the equation, -2.33(5.2) + 17.8 and ended up with 5.684. That is lower than the mean though which does not make sense to me, why would the shop guarantee they can complete the job in 6 minutes (rounding up) is the mean is 17.8. Where did I go wrong? Thanks in advance!

 

[Ray Vickson]

Are you sure you have the interpretation correct? The shop can say that no more than 1% of jobs will be finished in less than 5.68 minutes; that is, they can say "we guarantee we won't finish your job before 5.68 minutes". Wouldn't the shop want to say "we can guarantee your job won't take _longer than_ X minutes"? That would make X greater than 17.8, not less.

RGV

 

[Mark44]

Homework Statement

The mean is 17.8 and the standard deviation is 5.2. The shop does not want to give more than 1% of its customers free service, how long should the guarantee be? (cut the problem short, the rest was just story)

Homework Equations

I think the equation I would use is X = Z(standard deviation) + Mean

The Attempt at a Solution

I changed 1% into the Z score and got -2.33

The z score you want is +2.33.

and plugged that into the equation, -2.33(5.2) + 17.8 and ended up with 5.684. That is lower than the mean though which does not make sense to me, why would the shop guarantee they can complete the job in 6 minutes (rounding up) is the mean is 17.8. Where did I go wrong? Thanks in advance!

 

[ChrisBlack]

Are you sure you have the interpretation correct? The shop can say that no more than 1% of jobs will be finished in less than 5.68 minutes; that is, they can say "we guarantee we won't finish your job before 5.68 minutes". Wouldn't the shop want to say "we can guarantee your job won't take _longer than_ X minutes"? That would make X greater than 17.8, not less.

RGV

The problem says that the shop does not want to give more than 1% of customers a free oil change, so I would think it would be over the mean, which makes sense if the Z score is positive, but i don't see why it's positive. Sorry, I should have written the whole problem

 

[Mark44]

Right. So if the z-value for the time it takes the shop to do an oil change is larger than 2.33, that represents only 1% of the customers.

Now, you need to translate that z-value of 2.33 to an x-value, which represents the cutoff time for an oil change. If the shop takes longer than that cutoff value, the oil change is free.

 

[ChrisBlack]

The equation I posted at the top is the correct one to change the Z value to an X right? the answer i got it 29.91, or 30 minutes

 

[Mark44]

Yes, your equation is fine, and barring any arithmetic errors (I didn't check), your answer looks fine.