Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to recognize a balanced design in two-way ANOVA hypothesis testing.  Here's our problem statement: Researchers randomly select and weigh men and women.  Their weights are entered in the table below so that each cell includes five weights.  Is the result a balanced design?  Why or why not?

Well, here we have our samples here in the table.  And we notice we have the same number of samples for each of the cells in the table.  This is what it means when it's talking about balanced design.  So here we've got five samples, and here we've got five samples.  You don't have any more samples for any combination of row factor or column factor than you do for any other.  And that's what it means by balanced design.  So we see we have five samples for each of the cells in our table.  So that means that we do have a balanced design.  Excellent!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to perform the Kruskal-Wallis test for chest deceleration measurements.  Here's our problem statement: Use the following listed chest deceleration measurements (in g, where g is the force of gravity) from samples, small, midsize and large cars.  Use a 5% significance level to test the claim that the different size categories have the same median chest deceleration in the standard crash test.  Do the data suggest that larger cars are safer?

OK, the first part of this problem asks for the null and alternative hypothesis.  With the Kruskal-Wallis test, it's pretty much standard.  The null hypothesis is going to be that all of the means or median values are going to be the same.  And the alternative hypothesis is going to be that at least one of them is different, so they're not all the same.  And let's see what we got here.  So we want equal medians, not all equal.  This looks good.  Excellent!

Next we're asked to compute the test statistic.  StatCrunch makes this super easy.  So I'm going to dump my data here into StatCrunch.  Let's resize this window so we can see better what's going on.  OK, here in StatCrunch, I go to Stat --> Nonparametrics --> Kruskal-Wallis.  Here in my options window, I'm going to select my columns, and that's all there is to it.  Here's my test statistic.  I'm asked to round to three decimal places.  Excellent!

The next part asks for the P-value.  We've already got that calculated.  It's right next door to the test statistic here in the results window.  Were asked to round to four decimal places.  Good job!

And now we're asked to state our conclusion and answer this question: "Do the data suggest that larger cars are safer?"  Well, are we going to reject or fail to reject the null hypothesis?  Well, we've got a P-value of just under 5%.  Our significance level is 5%, so we're just inside the region of rejection.  But it doesn't matter whether you're in a little bit or way in; in is in.  So we're going to reject the null hypothesis, and we're going to say that there is sufficient evidence.  And what are we having sufficient evidence for?  Rejecting the claim samples are from populations with the same median, because that's what this says right up here.  We're rejecting the null hypothesis that says they all have the same median value.  Nice work!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to recognize the effect of data transformations on two-way ANOVA results.  Here's our problem statement: The accompanying data shows a sample of pulse rates in beats per minute that were categorized with two factors: age bracket in years and gender.  The data were used to illustrate the way to use a two-way ANOVA.  How are the results affected in each of the following cases? A) the same constant is added to each sample value; B) each sample value was multiplied by the same non-zero constant; C) the format of the table is transposed so that the row and column factors are interchanged; D) the first sample value in the first cell is changed so that it becomes an outlier.

OK, so the first part of this problem, Part A, wants us to consider if we transform the data by adding the same constant to each data value, how does that affect the ANOVA results?  Will the test statistic change?  Will the P-value change?  Well, if you consider the way that the test statistic is actually calculated, you come to understand that you know it's based on differences between individual data values and the mean value and then you square those differences.  So you know adding the same constant to each sample value is going to shift everything on your number line.  But the differences between the mean value and each individual value is still going to be the same.  So you know when you square those differences you're going to get the same numbers coming out.  So that's not going to change your results at all.  Nice work!

Now Part B wants us to consider a data transformation in which we multiply each sample value by the same non-zero constant.  Well, this again is going to not have an effect, because when you multiply each data value with the same non-zero constant, in essence what you're doing is you're keeping the proportions of the squares of the differences the same.  So again, that's not going to change anything.  Nice work!

Next, Part C wants us to consider if we transpose the row and column factor so that their data are interchanged.  So the row data becomes the column data, the column data becomes the row data.  Well again, you know the calculation for your values there in your ANOVA table.  And the division that you make to get this, the test statistic, it's kind of similar to what you see with the linear correlation coefficient where it doesn't matter if you swap the X and Ys; you're still going to get the same value out for R. It's the same sort of thing here.  You're still going to get the same values that you use to divide to get your test statistic out.  So the test statistic is going to stay the same.  And of course, if the test statistic stays the same, so is the P value; that's not going to change either.  So yeah, you can swap data around all you want.  That's not going to change anything.  Nice work!

And now the last part, Part D says, "Choose the correct answer if we just change the first data value in the first cell so that it becomes an outlier."  Well, OK, now we're going to see some changes here because, see, before in each of these three instances we were changing all of the data the same way.  Now we're going to change just one data value.  We'll leave the rest of it alone.  Now that's good.  Now we're going to see a difference with that.  And just as we saw with wide swings in our linear correlation coefficient with the introduction of an outlier in our data set, we're going to see the same thing here.  If you put an outlier in your data set, that's going to radically shift the value of the test statistic.  And hence also the P-value is going to change too.

So let's see what we got here.  Yeah, this one looks good.  "Both the test statistic and P-value will most likely change because outliers can dramatically affect and change the results of an ANOVA."  Yeah, that's going to be the one they want, but let's check the other options just to make sure.  "The P-value and mean will only change by a very small amount because ANOVA is robust against outliers."  That's definitely not true.  "The P-value will be approximately 1 minus the previous P-value."  Where in the world did that come from?  I don't know.  OK, so I'm pretty sure with the answer we got up here.  Nice work!

And that's the end of the problem.  And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below, and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to perform the runs test for randomness with law enforcement fatalities.  Here's our problem statement: Listed below from left to right and then top to bottom are numbers of law enforcement fatalities for 20 recent and consecutive years.  First, find the mean, identify each value as being above the mean (A) or below the mean (B).  Then test randomness above and below the mean using alpha equals 0.05.  Is there a trend?

OK, here we have our data.  And the first part of the problem is asking for the mean of these data.  So let's go ahead and dump that data into StatCrunch.  So here we have the data now loaded into StatCrunch.  So we're going to go to Stat --> Summary stats --> Columns.  Here in my options window, I select the column where my data can be found.  And then I want to select the mean.  Now I'm ready to go get it.  And here it is.  We're not asked to round to any number of digits.  In fact, the instructions specifically say do not round.  So I won't.  Nice work!

Now the next part wants us to determine the null and alternative hypotheses.  This is pretty much set for runs tests for randomness.  So here we're going to have an alternative --- excuse me, a null hypothesis that says the data are going to be in a random order.  And then of course the alternative hypothesis will be the alternative to that, which is that the data are in an order that's not random.  Good job!

Now the next part wants us to find the test statistic.  And to do this we need to figure out what our sample sizes are.  So let's go ahead and do the categorization that was mentioned here in the promise statement . So every one of these values that is above the mean, I'm going to categorize with an A.  And everyone that is below, I'm going to categorize with B.  So now I just need to go through and categorize each one of these values in turn.

Well, 158 is just barely above the mean and 157 just a little bit below the mean.  This is the kind of thing that a computer is really adept at doing . And that's why, you know, I wish that some of this functionality had been programmed into StatCrunch, because, I mean, it's really not that hard to do the --- I don't think it would be that hard to do, and you wouldn't have to do all this manual labor. I mean, come on, it's the 21st century.

OK, so here we've got all of our categorizations done.  Now we just got to do the counting.  So how many As do we have?  One, two, three, four, five, six, seven, eight, nine, ten As.  And we've got one, two, three, four, five, six, seven, eight, nine, ten Bs.  10 and 10.  So 10 is below 20, so both of our sample sizes are less than 20.  And we've got a 5% significance level here, so that means we can use the number of runs as our test statistic.  So how many runs do we have?  Well, let's find out here.  We've got one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve; I see 12 runs.  So that's going to be my test statistic.  Excellent!

Now we're asked to determine the P value and --- Oh, OK, here I just noticed something here.  I've got an error in my --- do I have an error?  It looks like, yeah, I do have an error.  163 is actually above the mean value, so I need to replace that with capital letter A.  OK, so that's going to change my counts to 9 and 11.  We're still --- both numbers are still less than 20, so we're good here with the test statistic, but now we got got to get the P-value.  And to get the P-value from StatCrunch, we first have to calculate our Z score test statistic.  And to do that we've got to do this manual hand calculation, the old school way looking at all of these.  Oh my gosh!  It's just a behemoth of an equation.

Anywho, let's get to it.  So let's see. What are our values? We got g, which is the number of runs; that's 12.  And then we've got our sample sizes, which are going to be 9 and 11.  Then we just substitute those values into our equation, and we simplify, and we punch into a calculator and out comes our Z score test statistic of 0.510807.  And the decimal just keeps going on and on and on and on and on and on.

So now I can take that number, come back into StatCrunch, and I'm going to select Stat --> Calculators --> Normal.  Here in my normal calculator, I want to select the Between option because we need a two tailed test.  And then here I just put in that Z score test statistic that I just calculated.  And I'm not going to take all those decimal places; four should suffice.  Whoops, it helps to put in the right number.  Then I just press Compute!, and out comes the area in between the tails.  The P-value is the area inside the tails, so I need to take that number, and I'm going to subtract it from one.  So one minus the area in between the tails gives us the area outside the tails.  And this --- or excuse me, the area of the tails, and this is our P-value right here, which we're asked to round the six decimal of, excuse me, three decimal places.  Nice work!

Now the next part wants us to determine a conclusion for our hypothesis test.  With such a high P-value, we're going to be well above our significance level.  So that means we're outside the reason of rejection.  Therefore, we failed to reject the null hypothesis.  Whenever we fail to reject the null hypothesis, there is not sufficient evidence.

But what is there not sufficient evidence of?  What is there not sufficient evidence for?  Well, we failed to reject the null hypothesis, so there's not sufficient evidence to side with the alternative hypothesis, which we see here is saying that the data are not in random order. So that's what we're going to put here.  They're in an order that's not random.  Fantastic!

And now the last part of this problem asks, "What do the results suggest?"  Well, if we can't conclude that the data are in a random order, then we're supposing that the data are in a random order.  That's what we --- that's why we failed to reject the null hypothesis, because it's potentially true.  It's potentially true the data are in a random order.  And so if there's a random order, that means there's no trending.  And if there's no trending, that means the values are scattered above and below the mean value.  Nice work!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below, and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.

Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to perform Spearman's rank correlation test for audience impressions.  Here's our problem statement: The table below lists the numbers of audience impressions in hundreds of millions listening to songs and the corresponding numbers of albums sold in hundreds of thousands.  Does it appear that album sales are affected very strongly by the number of audience impressions?  Use the significance level of alpha equals 0.05.

OK, the first part of this problem is asking for null and alternative hypotheses and for a rank correlation test.  That's pretty much going to be set.  The null hypothesis is going to say there is no correlation, so it's going to be equal to zero.  And the alternative will be that there is correlation, so the correlation coefficient will not be equal to zero.  We want to get a population parameter because we always use population parameters with our null and alternative hypotheses.  So we don't want to choose a sample statistic, and that means we're going to select this answer option here.  Nice work!

Now the next part asks us to find the value of the correlation coefficient.  To do that, we're going to take our data and dump it into StatCrunch.  OK, here we are in StatCrunch.  And now I'm going to go to Stat --> Nonparametrics --> Spearman's correlation.  Here in my options window, I'm going to select the columns where my data can be found, and I'm just going to hit Compute!, and here it is.  There's my correlation coefficient.  Notice the negative sign out in front.  We want to be sure to include that.  Fantastic!

Now the next part asks for critical values, and critical values are obtained from a table where we've got less than 30 for sample size.  And here we definitely have less than 30 for sample size.  So n is equal to 9.  So we've got nine sample pairs, and we're asked to use the significance level of 5%.  So if I look on the table here for sample size of 9 and 5%, I'm going to get 0.7.  So I've got two critical values, so I'm going to use my plus or minus sign so I only have to type the number in once.  Well done!

And now the last part asks us to resolve the hypothesis test.  Well, here we've got a test value --- a test statistic, rather, of -0.181, and that's going to be between 0.7 and -0.7.  So therefore we're outside the region of rejection, and we're going to fail to reject the null hypothesis.  And every time we fail to reject the null hypothesis, there's insufficient evidence.  But that's not how these are worded here.  So let's see what we got.  Yeah, this one says there appears to be a correlation between the number of audience impressions and the number of albums sold.  And this one says there does not appear to be a correlation.  So which is it?

Well, we failed to reject the null hypothesis, which means it's potentially true.  And here the null hypothesis says there is no correlation.  So it's potentially true that there is no correlation.  So there does not appear to be a correlation is what we're going to answer.  Well done!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below, and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to perform proportion hypothesis testing on vehicles with front license plates.  Here's our problem statement: Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, and then state the conclusion about the null hypothesis as well as the final conclusion that addresses the original claim.  Among 2072 passenger cars in a particular region, 241 had only rear license plates among 307 commercial trucks.  45 had only rear license plates.  A reasonable hypothesis is that commercial truck owners violate laws requiring front license plates at a higher rate than owners of passenger cars.  Use a 10% significance level to test that hypothesis.  A) Test the claim using a hypothesis test.  B) Test the claim by constructing an appropriate confidence interval.

OK, so Part A wants us to conduct a hypothesis test, and the first part of Part A asks us for the null and alternative hypotheses.  So here we're going to look at not selecting answer option D because the null hypothesis here is not a statement of equality.  Of the three answer options that remain, we need to look at the alternative hypothesis.  To do that, we go back and look at the claim that's being made.  And the claim is that commercial truck owners violate laws requiring front license plates at a higher rate than the owners of the passenger cars.  So trucks are going to be greater than cars, but which is 1and which is 2.  Well, if you read the problem statement, the cars are mentioned first, so they're going to be the first proportion.  And then the trucks are going to be the second proportion.  So trucks are greater than cars. That means 2 is greater than 1, which means 1 is less than 2.  And the alternative hypothesis that says 1 is less than 2 is going to be this one right here.  Well done!

Now the next part of Part A wants us to identify the test statistic.  And to do that, we need to whip out StatCrunch.  So I'm going to pop out StatCrunch here.  And we'll resize this window to give us a better view of what's going on.  OK, here in StatCrunch, I want to go to Stat --> Proportion stats (because we're dealing with proportions) --> Two sample (because we have two samples) --> With summary (because we don't have any actual data).

Here in the options window, we need to put in some statistics for our samples.  The first sample is the one that was mentioned first, which is the passenger cars.  So the number of successes --- we're going to consider a success having only a rear license plate, so I take that number right there from the problem statement --- 241.  And I'm going to put in the total number of observations, which is the total number of cars --- 2072.  I do the same thing with the trucks.  And now down here under Hypothesis test, I want to make sure that this matches what we have earlier for our null and alternative hypotheses.  Notice the format is written differently, but that's OK; they're algebraic equivalents.  If I just take an add p2 to each side, I get the same thing.  It's listed right over here.  So I want to make sure that symbol is the same as this symbol, and now it is.  And so I'm just going to leave that zero alone because that makes these two algebraic equivalents.

Now I'm ready to go and get my test statistic.  And here we see the test statistic right here, second to last value there in that results window table.  That's good.  I'm asked to round to two decimal places.  Nice work!

Next we're asked to identify the P-value.  The P-value is right next door to the test statistic; it's that last value there in that results window table.  And I'm asked to round to three decimal places.  Good job!

Now the fourth part of Part A says, "State the conclusion about the null hypothesis as well as the final conclusion that addresses the original claim."  Well, if I go back and compare my P-value with my significance level --- and let's see, where do we have our significance level?  I'm looking, I'm looking, I'm looking.  Wow, I don't see where it --- oh, it's right here.  Duh!  Right in front of you --- 10% significance level.  So if I come down here, I look at 10% significance level.  My P-value is 6%, so we're under the significance level, which means we're inside the region of rejection.  Therefore, we're going to reject the null hypothesis.  And whenever you reject the null hypothesis, there's always sufficient evidence.  Excellent!

Now Part B asks for a confidence interval from the same data.  So I could go back through all those motions again, but I'm lazy.  So I'm going to come back up here, click on Options --> Edit, and then down here I'm gonna switch this radio button to Confidence interval.  And they don't specify a confidence level, so we have to determine the appropriate one.  If we've got a 10% significance level, that would mean alpha is 10%, but we've got two samples.  So I've got to subtract two alpha, and that's going to give me an 80% confidence level.

Now I got my upper and lower limits for my confidence interval, and I can place those in here.  We're asked to round to four decimal places.  It's making me count today!  There's my lower limit.  Now I get to put in the upper limit.  Fantastic!

And now the second part of Part B asks us to interpret the confidence interval, which we can see here.  We look at their confidence interval.  It does not contain zero, and so, because it does not contain zero, that means one of these proportions is always going to be bigger than the other.  Since the entirety of the confidence interval is in the negative region of our number line, this difference is always going to be negative.  So that means p2 is always going to be greater than p1.  And that was the actual claim that we were making, because 2 corresponds with the trucks, 1 corresponds with the cars.  And so, 2 being greater than 1 means that the trucks are going to be greater than the cars, which means their rate of noncompliance is higher than the rate of the owners of the passenger cars.

So because the confidence interval does not contain zero, there is a significant difference between the two proportions.  Because there's a significant difference, that means we can reject the null hypothesis, because the null hypothesis is a statement of equality.  And every time we reject the null hypothesis, there is sufficient evidence.  Excellent!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below, and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.