"On-hold" times for callers to a local cable television company are known to be normally distributed with a standard deviation of 1.3 minutes. Find the average caller "on-hold" time if the company maintains that no more than $10 \%$ of callers have to wait more than 6 minutes.
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Okay, Well, we you know, we have some cable television company and they have their wholesome information about their hold times, and it's normally distributed. So let's on our diagram label out what we know. So they kind of analyze what we need to find out for this problem. So it's labeled this standard deviation is 1.3 minutes, so that's kind of a variation of the whole time there. So that's our standard deviation. What we don't know in this case is the mean So we want to find in this case, what is the meaning, it old time. So we must have one of the piece of information. So are other piece of information is the average caller, uh, usually wait to no more than no more than 10% Have to wait more than six minutes. So 10% gonna be over here, CASS, or our usefulness here is we know that 10% but this area of 10% uh, have to wait no more than six minutes. So we do know for Margraff, six minutes is over here. So what we're gonna do is going to use a calculator to work backwards, and it will use a little bit of Z score information because we know if you find disease four that goes with that with that number we know we know that the definition of the Z score is the number we know minus the mu. What about the standard deviation? So if we can find all the known parts, we can work backwards and find the unknown part. So let's used the inverse normal distribution. Since this is the normally distributed, um, I can use inverse Norman on my calculator. Well, from a table, he could work backwards from the table toe. Look at the part we know about. So ah, the 10% is going to school correspond to a zis for so an area have 10% meeting 10% or less of the whole times. In this case, we don't have the mean and standard deviation, so we can't put numbers in there, but we can get disease score that goes with that someone over here, right? My disease. Four of negative. 1.2 a is what a Z score that corresponds to the 10% there. So negative 1.28 is my Z score. That goes with my six minute whole time minus and mean whole time for that. And Senator Aviation for this is a 1.3. So what I want to do here, is it gonna multiply both sides by 1.3? So, Aiken soul for this guy here, So smoke level asides the equation by 1.3 under the 1.3. So what's gonna hit times? So we're gonna have a negative 1.6 repeating or make it a 1.66 not fully repeating, I guess equals 26 minus the mule. Just a symbol for me. And really, it adds the, um, you value to that side and then take the six. Plus the 1.66 to figure out the mu is So, um, that's going to do that. So let's take a six class corn army values gonna be seven minutes, 7.67 and 2/3 made it super using fraction world. So the mean hold time is 7.6. We'll see its about 7.6. About 7.7 minutes is the average hold time