00:01
So in this question, for the first part, we are going to assume here that because we don't know the value of this standard deviation in the population, so this is unknown, we need to use for all these items the t -distribution.
00:17
So the t -distribution has degrees of freedom equal to the sample size minus one, so in our case 58.
00:24
And now we are going to use the t -table to find the critical values to find each one of the confidence intervals.
00:30
So for the 90%, the critical value that we want has a middle area equal to 90%, and we want the upper level of this middle area in the t -distribution.
00:42
So if you use the t -distribution table, you're going to find that the value that has this middle area equals to 90%, when we are assuming that we have 58 degrees of freedom, is the value 1 .67.
00:58
But now let me just give some space here to find the confidence interval.
01:02
So in this example, we put the sample mean plus and minus this critical value times the standard deviation divided by the square root of the sample size.
01:12
In this case, when we compute this, we're going to get two bounds.
01:16
The lower bound here is the one that we get when we use the last sign, so in this case what we're going to get is 22 .5, or when you're rounding this to only one decimal place.
01:27
Then when we consider the plus sign, we get the upper bound, which will be, in this case, 24 .5.
01:36
So that will be the answer for a.
01:39
For b, the only difference is the critical value, because now the middle area is equal to 95%.
01:46
So when we have a 95%, the critical value here is equal to 2, and then we use the same formula...