0:00
All right.
00:01
We're tasked with estimating the true number of travel days for salespeople at a given company, which is going to be mew, at the 90 % confidence level.
00:14
And we want to estimate this within two days.
00:20
And we're told that a pilot study gave us a sample mean of 150 with a sample standard deviation of 14.
00:28
And this number is going to be important to calculate the number of the sample size or the number of people.
00:35
That we need to project or to predict our true population mean of the number of travel days within two days with a 90 % confidence level.
00:54
And so if we want to predict our population mean within two days at the 95 % or at the 90 % confidence level, we need then that our mu minus x bar divided by our standard deviation divided by the square of n, we need this absolute value to be less than or equal to, or actually, we need this value, the absolute value of the difference to be less than or equal to 2.
01:32
And so how can we convert this into a z statistic? well, z is going to be equal to this divided by, the standard deviation over the square of n.
01:48
And so we can do the same thing to the other side, which means that our z statistic, the absolute value, is going to be less than or equal to 2 divided by 14 divided by the square root of n.
02:11
And we also know that we want the probability, the probability of this occurring to be equal to 0 .90, or as close to 0 .90 as possible.
02:25
And so at what value of z are we going to get, or at what value of n and z is this equation going to be satisfied? so first, let's figure out what is the probability that the absolute value of z is less than a certain value.
02:45
What value of z does this occur for which the probability is 90? so if this is our z distribution, we want the probability that z falls in the, this range to be equal to 0 .90 or 0 .90, which means the probability on the positive end that z is less than this value should be equal to 0 .95.
03:13
That's because 0 .05 of the masses here, 0 .05 of the masses here, which means that we add these two together...