For each of the following situations find the critical...

For each of the following situations, find the critical value(s) for z or t. a) H0: p = 0.8 vs. HA: p ≠ 0.8 at α = 0.01 b) H0: p =0.2 vs. HA: p > 0.2 at α = 0.05 c) H0: μ =30 vs. HA: μ ≠ 30 at α = 0.05; n = 24 d) H0: p = 0.8 vs. HA: p > 0.8 at α = 0.01; n = 345 e) H0: μ = 40 vs. HA: μ < 40 at α = 0.05; n = 1000 a) The critical value(s) is(are) Z or T = ? (Use a comma to separate answers as needed. Round to two decimal places as needed.) b) The critical value(s) is(are) Z or T = ? (Use a comma to separate answers as needed. Round to two decimal places as needed.) c) The critical value(s) is(are) Z or T = ? (Use a comma to separate answers as needed. Round to two decimal places as needed.) d) The critical value(s) is(are) Z or T = ? Use a comma to separate answers as needed. Round to two decimal places as needed.) e) The critical value(s) is(are) Z or T = ? (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Transcript

00:01 So for each one of these situations, we need to compute the critical value of values.

00:06 And also say if you are going to use the standard normal distribution or a t student distribution.

00:12 So in letter a, we are working with proportions.

00:15 So proportions for sure, we are always going to use the standard normal distribution.

00:21 So you are going to have z critical values.

00:24 So in this case, for the first one, we have this hypothesis.

00:28 And we are considering alpha equals to 0 .01.

00:32 So first what we need to check, this test here is two -tail.

00:38 Why two -tailed or two -sided? because we have the different sign here.

00:43 When a test is two -tale, this means that we have two critical values.

00:50 And when we have one -sided test, which means that the alternative is less, in this case, less or greater.

01:01 This means that we have just one.

01:03 But in the first letter a, we have two.

01:06 And then to find the critical values, like i said, for proportions, we always use z.

01:12 Okay? so we're going to have like z1 and z2.

01:17 But they are the same.

01:18 The only difference is because one will be the positive and the other one will be the negative.

01:23 But the value is the same.

01:25 And how can we find this value? we are going to use this alpha here that we have, this value.

01:32 So basically, the value that we want is the value in the standard normal distribution.

01:37 So i'm going to put here z1 because this is the positive.

01:42 There has, in this case, since alpha is 0 .01, this means that this area here, because this test is two -sided, this means that we have alpha divided by 2.

01:54 The 2 is because the test is 2 -sided.

01:56 So this here is the same as 0 .005.

02:00 So the area left to this number is 1 minus this number.

02:05 So in this case, 0 .995.

02:08 Now assuming this value here, we can use the z table to find that the positive here is 2 .576 and the negative is just the negative of the same number.

02:20 Now in letter b, again, we have proportions.

02:24 So we are also going to use the normal distribution.

02:29 So now what we need to do is basically check if the task is two -sided or not.

02:35 So in our case, now the task is one -sided.

02:38 Why? because we don't have the difference.

02:40 We have the greater sign.

02:42 And this greatest sign will give us if you are going to find the value in the right side of the normal distribution or not.

02:53 So because this is greater, this means that if it's a value, if i draw here, my number or my critical value will be here, like in the right side.

03:02 And the area here, i don't need to divide by two because it is one -sided.

03:06 So we just have one critical value.

03:09 So i can put all the alpha that i have in this side, which means that i have 0 .95 left to this number.

03:18 And using the z table, you're going to find that this here is 1 .640.

03:26 Now in letter c, we have a, in this case, a test for the population mean.

03:34 So now for the population mean, what we need to check is basically if the sample mean, the sample size, sorry, is greater or not than 30...