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We finally reached the end of our hypothesis testing procedure.

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And now we're going to go through the last few steps where we get to determine whether or not our result

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is significant enough to lend support to our alternative hypothesis.

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So to pick up where we left off, we started out the hypothesis testing procedure by stating null and

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alternative hypotheses.

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And along with that we chose a confidence level and therefore an associated alpha value.

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And we decided whether we wanted to run a one tailed test or a two tailed test.

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And then in the last lecture we learned how to calculate our test statistic and we calculated this t

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score because we always need to use a t test when population standard deviation is unknown.

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Once we have our T score or our t test statistic in hand, we need to determine whether or not that

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score is significant enough to reject the null hypothesis, and we can determine that significance in

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one of two ways.

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We can do it by looking at the p value, or we can do it by looking at the critical value.

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So we have two options here.

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If we use this p value method, then we're going to follow just three steps here.

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So we're going to locate the T score our T test statistic in the T table, and we'll do that by navigating

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to the appropriate row based on degrees of freedom and then scanning across the row until we locate

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the T score.

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Once we find the cell in the T table where we're positioning ourselves, we'll look at the corresponding

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upper tail probability at the top of the t table.

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And then based on the kind of test that we're running, we'll use that upper tail probability to determine

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p value and then we'll compare that p value to the alpha value that we set out at the beginning when

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we picked a confidence level.

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And we'll go through these steps in detail with an example.

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Otherwise, if we use the critical value approach, then we're following just two steps.

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This time we use our degrees of freedom, remember, based on sample size and the confidence level to

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locate a critical value in the body of the T table.

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And then we compare that critical value to the absolute value of the T score or the T test statistic

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that we calculated earlier.

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And then based on what we find, here are the important conclusions that we can make.

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If we use the p value approach, then if P is less than or equal to alpha, we reject the null hypothesis

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and therefore lend support to the alternative hypothesis.

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If we use the critical value approach, if the absolute value of t our t score we calculated when we

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found the test statistic, if the absolute value of T is greater than the critical value that we get

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from the body of the T table, then we reject the null hypothesis and therefore lend support to the

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alternative hypothesis.

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So that's a lot to take in.

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It'll be much easier to understand if we go through an example.

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So let's pull up here a T table and I've consolidated this table.

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So we're just showing the rows four, 18, 19 and 20 degrees of freedom instead of looking at the entire

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t table, because we're going to go through an example here where we choose a confidence level of 95%,

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therefore an alpha value of 5% or 0.05, and we'll run a two tailed test.

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And more specifically, we're going to say that we use a 20% sample or that our sample size is 20.

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So because our sample size is 20, remember then that degrees of freedom is always the sample size minus

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one.

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In our case, that's 20 minus one or 19.

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So that's why we're just looking at this section of the T table, because we're going to be looking

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at this second row here where we have 19 degrees of freedom.

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So to go back to the beginning here, what we're saying is that we are running a two tailed test.

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We decided to run a two tailed test because we're not sure about the directionality of our hypothesis

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statements, which means by definition, that the null hypothesis has to be that the population mean

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is equal to our hypothesized value.

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And we're going to say that the hypothesized value here is 14.

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That means that our alternative hypothesis is that the population mean is not equal to 14.

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In the last few lectures we've been using this hypothesized value where we say that we're running the

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shipping department in a warehouse and we're investigating whether mean order processing time is 14

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hours.

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Let's say that the assumption is that mean order processing time is 14 hours.

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We're going to run a two tailed test to investigate our alternative hypothesis here.

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That mean order processing time is not 14 hours.

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So we have these hypothesis statements.

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We can see from the hypothesis statements that we're running a two tailed test because of the fact that

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we have the equal sign and the not equal to sign.

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Otherwise, if the alternative hypothesis was mu greater than 14 or MU less than 14, then that would

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indicate a directional test and therefore a one tailed test.

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But we have these hypothesis statements.

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We're running a two tailed test.

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We've chosen a confidence level of 95%.

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That means that the alpha value is 100%, -95% or 5%.

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We can also call that alpha value 0.05 if we convert to decimal form there.

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And we're saying that we have taken a sample of 20 orders in our warehouse, we computed a sample mean

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of 14.4 5 hours with a sample standard deviation of one hour.

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And so our next step then is to calculate the t test statistic.

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And so we have our formula.

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We looked at this in the last lecture.

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If we plug in everything that we have here, the sample mean we said was 14.4 5 hours.

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We subtract the hypothesized value 14 hours, and then we divide that by sample standard deviation,

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which is one hour divided by the square root of the sample size.

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And in our case, the sample size here is 20 orders.

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Now, if we use a calculator to find this value, we get a t test statistic of approximately 2.012 when

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we round to three decimal places.

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So at this point we have taken care of all three of these steps here.

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And the whole first part of our hypothesis testing procedure.

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And now we need to either use the p value approach or the critical value approach to make a conclusion

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about whether or not this t score is significant enough to allow us to reject the null hypothesis.

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So let's look at the p value approach.

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If we use this approach, here's what we do.

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Once we have our test statistic, we locate this test statistic in our t table and we do that based

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on degrees of freedom.

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So in this case, with a sample size of 20 degrees of freedom is 20 minus one or 19.

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So we go to the row where we find 19 degrees of freedom and then we start moving our way across the

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row looking for 2.012.

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Now, keep in mind that it's very, very unlikely that we'll find exactly this value for the T score

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that we calculated.

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Almost certainly we'll find a T value here that lies in between two of the values in the table.

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So looking across this 19 degrees of Freedom Row, we see that this score right here lies in between

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these two values.

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It's in between 1.729 and 2.093.

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So since we're in between these two values and our next step is to locate the corresponding upper tail

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probability, we want to look at the upper tail probability associated with both of these values, the

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value to the left and right of the approximate location of our T score.

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So if we go up to the top of our T table, we can see the upper tail probabilities associated with those

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two values here.

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0.05 and 0.025.

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Now, usually what we'll do at this point, because we know that our T score is somewhere in between

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these two values, we'll pick some value in between these two upper tail probability values, some value

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between 0.05 and 0.025.

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So let's just say for the sake of argument that we split the difference equally.

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In other words, we take the halfway point between 0.05 and 0.025.

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Well, the value exactly halfway between these two is 0.0375.

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Now, once we found this value here, if we are running a two tailed test, which is what we're doing

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here, we need to multiply this value by two.

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So we would take two times 0.0375 and we get 0.075.

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And this then is the P value.

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The P value is equal to 0.075.

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Now here's the reason that we multiply that value by two.

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When you hear p value think probability value.

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The value of P is essentially giving us the probability of rejecting the null hypothesis.

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Or another way to put that would be the probability of falling within the region of rejection.

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And another way to put that would be that it represents the amount of area under the distribution that

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makes up the region of rejection.

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And that's normally what we imagine Alpha to be, which means that if we get a P value that is smaller

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than the alpha value, it means that we're going to fall within the region of rejection and therefore

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that we can reject the null hypothesis.

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But if we get a p value that is larger than the alpha value, it means we're falling outside the region

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of rejection and therefore that we won't be able to reject the null hypothesis, will fail to reject

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it, which is why we can compare the p value directly to the alpha value.

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We want to look at them in direct comparison and that's why we double this value that we found here

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when we're running a two tailed test.

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Because from the T table we just get an upper tail probability value.

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And so this estimate here is an.

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Estimate of the amount of area we'd find in the region of rejection in the upper tail.

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But if we're running a two tailed test, we expect this same amount of area in the lower tail.

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And so to find P for a two tailed test, we have to double this value to capture the full region of

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rejection from both tails.

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So for a two tailed test, we double this value that we find and then we compare that double value directly

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to Alpha.

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In this case, alpha is 0.05 and P is 0.075.

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So within this particular set of circumstances, P is greater than alpha, which means that we'll fail

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to reject the null we can't reject.

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The null hypothesis.

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If instead of using the p value approach, we use the critical value approach, then our first step

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is to locate the critical value in our t table.

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What that means is that we look up degrees of freedom and our confidence level.

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So here for this example, we know we're at 19 degrees of freedom at a confidence level of 95%.

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So we come to 19 degrees of freedom and move across until we see that confidence level of 95%.

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And that puts us here at this critical value.

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We call this the critical value when we find the intersection of those two things, 2.093.

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So 2.093 is our critical value and we want to compare the absolute value of the T score that we calculated.

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Remember that absolute value just means taking the positive version of whatever value we have.

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So even if here we had calculated RT is -2.01 to the absolute value of -2.012 is just positive 2.01

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to the absolute value of a positive number is itself.

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So all this means here to take the absolute value of RT is if we come up with a negative value for t

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taking the absolute value just removes that negative sign and turns the negative number into a positive

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number.

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Positive numbers stay the same.

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So here we have a positive number.

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Taking the absolute value means it stays the same.

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So we're comparing positive 2.012 to the critical value of 2.093.

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What we're showing here is that the absolute value of T is the absolute value of 2.012 compared to the

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critical value of 2.093.

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Well, that is less than 2.093.

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And because our RT score is less than our critical value, we fail to reject.

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We can't reject the null hypothesis.

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We need our RT score, the absolute value of our RT score to be greater than the critical value in order

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to reject the null hypothesis.

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So then in this scenario, the conclusion of our hypothesis test is to say that based on the sample

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values we calculated, the sample mean in sample standard deviation at a sample size of 20 orders and

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to be 95% confident running a two tailed test.

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We fail to reject the null hypothesis.

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We can't reject the null, which means that we can't lend support to our alternative hypothesis.

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There's no evidence here or there's not enough evidence in this result to support the idea that mean

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order processing time is something other than 14 hours.

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And that would then be the conclusion of this hypothesis test.

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But let's look at what happens if we run a one tailed test instead of a two tailed test.

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So we keep the same confidence level and therefore alpha value.

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We want to be 95% confident, but instead this time we're running a one tailed test, which means therefore

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that we have to change our hypothesis statements.

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We'll say that we're going to hypothesize that mean order processing time is greater than 14 hours.

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Therefore, the null hypothesis would need to say that mean order processing time is not greater than

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meaning less than or equal to 14 hours.

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So we change our hypothesis statements, but we still took a 20 order sample.

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We found a sample mean of 14.4 5 hours, a sample standard deviation of one hour.

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And of course, this is still our hypothesized value, which means we're going to compute the same t

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score, the same T test statistic of 2.012.

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Now let's go back to our P value approach.

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But first, when we look here at our t table, we need to be careful because this t table is really

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set up for two tailed tests.

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If you can see here the confidence level across the bottom.

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Let's take this 90% confidence level here.

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That means the alpha value would be 10% for this confidence level, which means that if we're running

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a two tailed test will have 5% of the alpha value, that 10% alpha value will be divided between 5%

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in the lower tail and 5% in the upper tail.

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And you see here the upper tail probability that corresponds to 90% is a 5% upper tail probability,

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which means that this t table by design is representing a two tailed test because this upper tail probability,

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if we match it up to the confidence level by definition is showing that upper tail probability for a

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two tailed test.

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So we need to be aware of that, that this row here, this confidence level row that's given to us by

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default is for a two tailed test.

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If we're running a one tailed test, notice that these confidence levels shift over one column to the

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left while upper tail probability stays the same.

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So for that same 90% confidence level that was here, if we have a two tailed test, we know that that

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10% alpha value gets divided into 5% in the lower tail and 5% in the upper tail, which we see right

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there, but for a one tailed test.

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That means the entire 10% alpha value is going to be put into one tail alone.

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So we shift these confidence levels over one column to the left so that this confidence level of 90%

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corresponds to the upper tail probability of 10%, because the entire 10% alpha value that's associated

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with this 90% confidence level is pushed into one tail.

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So we need to see 90% confidence with 10% alpha in a one tailed test versus 90% confidence with 5% alpha

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in a two tailed test, which means then that if we're going to use this p value approach and follow

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these steps again, if we again locate our T score in the T table.

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So we calculated a T score of 2.012 and based on the degrees of freedom of 19, we locate that spot

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in this row.

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And again, just like before we find this spot right here in between 1.729 and 2.093, we locate the

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corresponding upper tail probability.

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So we come up here to the top because we're in between these two values.

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Where in between these two upper tail probability values, 0.05 and 0.025.

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So if we then pick a value that's exactly halfway between those two, we pick 0.0375.

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That's exactly halfway between 0.05 and 0.025.

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But because we're only running a one tailed test, we do not double this value like we did before when

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we were running a two tailed test.

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Now we keep the p value as just 0.0375.

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So this time we get P equals 0.0375.

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We compare that to our alpha value of 5% or 0.05.

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And we see here in this case that P is less than alpha.

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And based on that finding, we reject the null hypothesis.

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So this is here the two tailed.

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Scenario, and this is the one tailed scenario.

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So in the one tailed scenario based on the p value approach, we're able to reject the null.

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What about using the critical value approach with a one tailed test?

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Well, here we again locate the critical value in the T table based on degrees of freedom and confidence

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level.

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So four degrees of freedom 19 and a confidence level of 95% using our one tailed test row for confidence

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level.

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So 19 degrees of freedom over to 95% confidence.

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And we arrive here at 1.7 to 9.

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So our critical value this time, remember last time it was 2.093, Now it is 1.7.

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Two nine and let's put this one here in white to indicate that it was part of our two tailed test.

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So for the one tailed test, we get 1.7 to 9 is the critical value we need to compare now the absolute

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value of 2.012.

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So absolute value of 2.012.

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And we need to compare that to the critical value of 1.729.

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Well, in this case, 2.012 is greater than 1.729.

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And because we get this greater than value, we can again reject the null hypothesis, which means that

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if we run this as a one tailed test, then we are successful at rejecting the null and therefore successful

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at lending support to the alternative hypothesis.

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So in this case we would say that we've found evidence for the idea that mean order processing time

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is longer than 14 hours.

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The conclusion of this particular hypothesis test with this particular sample that we found supports

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that result, supports that conclusion, that mean order processing time is more than 14 hours.

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So under the exact same set of circumstances, we failed to reject the null for the two tailed tests

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and succeeded at rejecting the null for a one tailed test.

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Remember, all the way back at the beginning when we talked about hypothesis statements and one and

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two tailed tests that we said that the two tailed test was more conservative and that we should only

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use a one tailed test when we had some kind of data, some kind of evidence to give us a reasonable

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belief about the directionality of the test.

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And that's because with the entire region of rejection shifted into one tail of the distribution, it

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makes it easier for us to clear that bar and move out of the region of acceptance and into the region

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of rejection.

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And we are more likely to reject the null with a one tailed test.

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The two tailed test is more conservative, and if we don't have a good reason, some strong suspicion

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to expect the directionality of the test, we should use the two tailed test since it is more conservative

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and harder for us to reject the null hypothesis, it's usually better to run a two tailed test and fail

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to reject the null or find no significance than it is to run a one tailed test and reject the null when

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we actually shouldn't, especially if the stakes are high like we talked about before.

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Especially if our hypothesis testing procedure relates to something like health and safety and we need

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to be really, really sure of our results.

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We want to bump up this confidence level as high as we can and make sure that we don't make a wrong

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decision here, especially in a scenario like that one.

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It's better to be more conservative and run this two tailed test unless we really, really have a good

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reason to state a specific direction in our hypothesis statements and run a one tailed test.

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But either way, that is the conclusion of the hypothesis testing procedure.

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This is the process we'll follow over and over and over again in inferential statistics.

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And it always ends this way where we calculate our test statistic and then we use either the p value

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approach or the critical value approach to determine whether or not that statistic is significant enough

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to allow us to reject the null hypothesis and therefore lend support to our alternative hypothesis.