In Exercise three, we're going to be considering a random sample that is drawn from a normally distributed population and this time the standard deviation is unknown and we've been to Use the given information to construct a 99% confidence interval for the population mean? So we have two cases in both cases, the sample uh mean and the sample standard deviation are 386 and 24 respectively. For the first case the sample size is 18 and the second cases sample size seven, So it's a smaller example for the second case, no first things. The first thing we need to do here is to determine the level of significance And also to determine the right formula that we need to use to get the 99% confidence interval. Now, since the uh population, standard division is unknown, we're going to use the following formula X. Bar plus or minus The critical value of tea for two tailed tests, multiplied by s example, standard deviation divided by the square root of n. And since it's a 99% confidence interval, it means that uh the value Of Alpha, the level of significance equals 1 -0.99 And that equals 0.01 Half of that is 0.005. So this is useful for us to determine the correct critical bottle 40. Now, if you're using the t distribution, you also need to give the degrees of freedom for each case. Now, for the first case, Because the sample size is 18 So that the number of degrees of freedom will be N -1, which equals 18 -1 and that equals 17. So you want to check for the corresponding critical value of T Where they for 0.0 Uh five as a level of significance. So that corresponds to the value two point eight 98. And now that we have all we need we can substitute um all the material and all the content that we had uh into the formula. So let's begin. So X bomb is 386 plus or minus. The critical value 40 is 2.898, Multiplied by S which is 24 Divided by the Square Root of N., which is square root of 18. And when you saw that the calculator you can to obtain 386 plus or minus 16.39. So the margin of error for this case is 16.39. Now let's proceed to the second case, We want to start by getting the degrees of freedom which will be N -1. And in this case it's going to be 7 -1 and that equal six. And so the critical value of T. There corresponds to this level of significant European 005 Is uh is obtained from the table, it is 3.70 uh 7. Then We can substitute our values into the formula again and you're going to get 386 plus or minus 3.707, multiplied by 24, divided by the Square Root of seven. and when you work that out, you're going to get 386 plus or Uh minus 33 0.63. So for the second case where we have a smaller sample, you can notice that the margin of error is greater, meaning that we're going to have a longer interval is compared to when the sample size is smaller.