00:01
Hi everyone, welcome to the problem.
00:02
So in this problem, they have given us that n is equal to 16 and x is samples from a population that is approximately normally distributed.
00:11
So x values are given and we have to find 90 percentage confidence interval for the population mean.
00:18
So coming to the problem first, the formula for population standard deviation is population standard deviation.
00:34
The formula is sigma is equal to square root of 1 divided by n.
00:42
Summation of i is equal to 1 to n x i minus mu the whole power square.
00:51
Okay so let's take this to be our equation 1.
00:56
So first let's find the value for mu.
00:59
So mu here the formula is summation of x divided by n.
01:06
So here are n from the question we already know that it is.
01:14
And from the question that we already know is 16, right? so it is 16.
01:21
So we will be writing mu is equal to, that is new.
01:28
So new is equal to 61 plus 85 plus 92 plus 77 plus 83 plus 81, plus 75 plus 78 plus 95 plus 87 plus 69 plus 74 plus 86 plus 84 so these are the x values okay so divided by the n value is 16 so we will be getting which is nothing but our mu is 80.
02:17
So this is our mean.
02:19
Now next we are going to find in summation of i is equal to 1 n x i minus mu the whole square that is x i the value first x value 61 minus our mean that is 80 whole square plus next the x value for the next x value 85 minus 80 the whole square plus the next x value 95 minus a mean the whole square plus 77 minus 80 the whole square plus 83 minus 80 the whole square plus 81 minus 80 the whole square plus 75 minus 80 the whole square plus 78 minus 80 the whole square plus 78 minus 80 the whole square plus 78 minus 80 the whole square plus 78 minus 80 the whole square plus 78 minus 80 the whole square plus 75 minus 80 the whole square plus 78 minus 80 the whole square plus 95 minus 80 the whole square plus 87 minus 80 the whole square plus 69 minus 80 the whole square plus 74 minus 80 the whole square so next 76 plus 76 minus 80 the whole square plus 84 minus 80 the whole square plus 80 minus 80 the whole square plus 83 minus 80 the whole square so when we find these values we will be getting 361 plus 25 plus 144 plus 9 plus 9 plus 1 plus 25 plus 25 plus 225 plus 49 plus 101 21 plus 36 plus 16 plus 16 plus 0 plus 9 so we will be getting the answer for summation of i is equal to 1 to n x i minus mu the whole square is 1 .1050 now from equation 1 so from 1 if you substitute it a sigma value will be square.
04:43
Square root of 1 by 16 multiplied by 1050.
04:49
That is equal to square root of 65 .625 .625.
04:56
That is equal to 8 .109.
04:59
So this is our sigma value.
05:02
Sigma is 8 .109.
05:05
Now we have to subtract one from the sample space, sample size n to find the degrees of freedom.
05:13
Okay.
05:14
So degrees of freedom, that is nothing but d .f is equal to subtract one from the sample size.
05:27
So it is 15.
05:29
So our df is equal to 15.
05:32
Next, we have to convert that 90 percentage of that confidence interval we have to find.
05:37
No, that 90 percentage we have to convert it into decimal...