Howdy!  I’m Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we’re going to learn how to find the coefficient of determination and the explained variation.  Here’s our problem statement: Use the value of the linear correlation coefficient R to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

OK, so the first part asks us for R-squared.  Even if you don’t know what the coefficient of determination is, you can get the value that you need to put in the answer field just by noticing they give you a value here for R and they’re asking for R-squared.  So all we have to do is square this value for R, and that gives us the coefficient of determination.

So let’s get out our calculator, and I’ll put in the value there for R, square it, and I’m asked to round to four decimal places.  Excellent!

And now the second part of the problem asks for “the percentage of the total variation that can be explained by the linear relationship between the two variables.”  This is the explained variation.  And the explained variation is exactly the same as the coefficient of determination; the two are identical.

So all I have to do is take this number here that I’ve already calculated in the first part and notice that it wants it in percent form.  So I just take that number and put it in percent form.  Good job!

And that's how we do it at Aspire Mountain Academy.  Feel free 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 just boring or doesn't want to help you learn stats, then 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 use StatCrunch to find a regression line equation from a given scatterplot.  Here’s our problem statement: Using the pairs of values for all 10 points, find the equation of the regression line.  After removing the point with coordinates (2,9), use the pairs of values for the remaining 9 points and find the equation of the regression line.  Compare the results from Parts A and B.

OK, Part A asks us for the regression line for all 10 points that are shown on the scatterplot.  Let’s blow this scatterplot up a little bit so we can see a little bit better what we’re dealing with here.  Notice there’s no icon here to actually dump the data into StatCrunch.  So I’m going to call up StatCrunch, and we’re going to have to put the ordered pairs for each of these data points in manually.

To start off with, I’m actually going to label my columns X and Y.  It makes a later portion of solving this particular problem a little more easy.  And now I’m going to go and get the ordered pairs for each of my data points.  So starting here with this point all by itself, the X value for that is 2, and the Y value is 9, so that’s (2,9).  Then for the points here that are in a square formation, we’ve got three points that have an X coordinate of 4, three points that have an X coordinate of 5, and three that have an X coordinate of 6.  And then we’ve got three points — these three points have Y values of 2, 3, and 4, as do the next three and the last three.  So there’s the ordered pairs for all 10 data points here in my scatterplot.

Now I’m ready to make my regression line.  To do that, I go to Stat –> Regression –> Simple Linear.  Here’s where labeling those columns becomes very useful.  The X variable is the X.  The Y variable is the Y.  And that’s all I really need to get the actual regression equation, so I go ahead and hit Compute!  And let’s expand this window so we can see better what’s going on.

So we see the actual regression equation here.  But notice how the answer fields are just asking for the coefficients of the regression equation.  I can take them from here, but there’s a lot of business going on here.  And it’s actually easier for me to see those values if I go down here to my parameter estimates table.  So see the 9 that is here originally?  That same 9 is here.  And then that’s your y-intercept.  And then the slope — the slope value here is the same as the slope value down here.  So I actually prefer to get my values for my regression line down here in the parameter estimates table.

So the first value here is my intercept.  And the next value — notice I’m carrying the negative sign with me.  Good job!

Now, Part B asks for the regression line equation for the set of nine points.  So in the problem statement, it asks us to remove this one point that is an outlier and get the regression equation for just these nine.  To do that, first I’m going to clear out this results window.  And then I’m going to come up to this row and select it.  And then his little arrow next to it gives me a drop down menu, and I’m going to select Delete row.

Now I’m going to bring back my results window.  Select Options –> Edit, and then I don’t need to change any of the settings, so I hit Compute! And it recalculates everything with the new columns that I have after I deleted that data point.  So now we can see we’ve got the values here.  So now my regression equation is y-hat equals 3.  Of course, zero slope means there is no X variable for my equation.  Well done!

And now Part C is the last part of the problem, and it asks us to “choose the correct description of the results below.”  We’ve got four options here, so let’s look at them one at a time.

Option A says, “The regression line is very similar in both cases.”  Ah, well, that’s not true.  I mean, just look at the regression line equations here.  This is a line with a negative slope to it.  This is a line with — well, there’s no slope to it because it’s just a straight, horizontal line.  So answer option A isn’t going to work for us.

Answer option B says, “The regression line changes, but the change is small.”  Hello?  You go from having a negative slope to having no slope at all.  And, you know, your y-intercept changes quite a bit, so I wouldn’t call that change small. So answer option B isn’t going to work for us.

Answer option C says, “The removal of the point has a significant effect on the regression line.”  Well, that’s true.  But let’s check out answer option D before we select this answer.

Answer option D says, “There is no regression line for the second case because the data are in a pattern.”  Well, there actually is a regression line; we’ve got the equation right here.  So answer option D isn’t going to work for us.  Answer option C is the one we want.  Good job!

And that's how we do it at Aspire Mountain Academy.  Feel free 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 just boring or doesn't want to help you learn stats, then go to aspiremountainacademy.com, where you can find out more about accessing our lecture videos or leave a comment to find out, you know, like what you actually want to see us provide for you in the future.  And thanks for watching!  And I hope this helped you.  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 linear correlation hypothesis testing for bill totals and tip amounts.  Here's our problem statement: Listed below are amounts of bills for dinner and the amounts of the tips that were left.  Construct a scatterplot, find the value of the linear correlation coefficient R, and find the P-value of R.  Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables.  Use a significance level of α = 0.01.  If everyone were to tip at the same percentage, what should be the value of R?

OK, the first part asks me to construct a scatterplot.  To do that, I’m going to take my data that we see right here, and I’m going to dump it into StatCrunch.  Now my data is here in StatCrunch, and I’m going to resize this window so we can see a little bit better everything that's going to go on.

OK, to make a scatterplot, I could just go to Graph –> Scatter Plot.  But I know I'm gonna have to do some linear correlation work to get the rest of the problem out.  And I also know I get a scatterplot when I go and do that linear correlation in StatCrunch.  So instead of going to Graph, I'm going to come up here to Stat –> Regression –> Simple Linear.  Here in my options window, I have to select the data for my X and Y variables.  The X variable is typically the one that’s listed first; in this case, it's going to be the bill, and the Y will then be the tip.

Then I want to conduct the hypothesis tests because there is a claim of linear correlation that we’re testing.  Notice how we have two hypothesis tests that appear, one for the intercept and one for the slope.  We don't really want this.  What we want is a test on the linear correlation, which for the population parameter for that will be the Greek letter rho.  That’s not we see here, but we're going to leave these defaults alone because these are the default settings that are here inside StatCrunch and they match the settings that we want to test for linear correlation.  It could be positive correlation.  It could be negative correlation.  We don't really know.  It's just going to be a test either way.

So we’re going to have the null hypothesis of course will equal zero.  The alternative hypothesis will be not equal to zero.  And that's what we want.  So we come down here and hit Compute!  And then I’m going to resize my results window.  And notice here how it says up here at the top “1 of 2.”  That’s because we’re looking at the first of two pages for our results.  The scatterplot is on the second page, and to get there I just press this little arrow button in the bottom right.  And lo and behold, here's my scatterplot.

Now in order to make comparison with my answer options much easier, I'm going to change the axis values on my scatterplot.  Notice here the Y goes from about 4 to about 16 where here in our answer options the Y goes from 0 to 25.  There’s also a similar disparity between the upper and lower limits that we see here on our x-axis.  So I going to go ahead and change that by clicking on this little icon with the three lines down in the lower left corner.  And then I click on X-axis, and now I can change my x-axis to match the minimum and maximum bounds that we see in my answer options.  I do the same thing for the Y.  And see, now our axes are the same, we’re actually comparing apples with apples, and so it’s easier to match what we have here in StatCrunch with the correct answer.

So let’s go through and look at these answer options one at a time to see which is the correct answer.  I start here with answer option A.  And this looks like it might be a possibility here.  Looking at this, this is about 35 to about 5, and 35 to about 5, and these points are coming in about the same way.  So A looks like a possibility, but let's look at the others just to make sure.

Answer option B?  Definitely not; this cluster of points right here is not represented here in our scatterplot.  So that’s not going to be it.  Answer option C?  Again, look at this.  If you imagine a line of best fit here; this line’s going to have a negative slope.  If you go from left to right, the points are generally trending downward.  Here we have a positive slope, so this is going to be right.  Answer option D?  Again, answer option D has a negative slope to it, so we’re not going to select that.  We’re goig to select answer option A.  Excellent!

Now the second part of the problem asks for the linear correlation coefficient.  We already have that here results window; I just flip back to the first page, and the value for R is located right here at the top.  So we’re asked to round to three decimal places.  Well done!

The third part of the problem asks me to determine the null and alternative hypotheses, which as I was saying before we were looking at the options window in StatCrunch, this pretty much set standard.  The hypotheses were testing in StatCrunch aren’t what we see here with the rho, but it's pretty standard when you're testing for linear correlation.  The null hypothesis in going to be equal to zero.   The alternative hypothesis will be not equal to zero.  Nice work!

Now the next part of the problem asks for the test statistic.  So that’s here in my results window.  You can see that we have the second to last number in our table is always a test statistic.  But here we've got two test statistics: We’ve got one for the intercept and one for the slope.  So which one do we want?  Generally, you’re going to want the one for the slope, since the slope has more of an influence on whether or not your line of best fit fits your data points than the actual y-intercept.  So I’m going to take this value here as the test statistic for my slope.  That's the test statistic that I want to put here in my answer field.  I’m asked to round to two decimal places.  Good job!

Next I'm asked for the P-value.  Again, I have two P-values here in my table, but I’m going to want the one here for the slope just like I did for the test statistic.  Here I’m asked to round to three decimal places.  Fantastic!

Now the next part asks me to compare the P-value with the significance level.  Here we have a P-value of a little over 7%.  My significance level alpha is 1%, so were definitely greater than the significance level.  And that means we are outside the region of rejection, and when you are outside the region projection, you fail to reject the null hypothesis.  And whenever you fail to reject the null hypothesis, there is not sufficient evidence.  Fantastic!

And now the last part of the problem asks if everyone were to tip with the same percentage, then what is the value of R?  Well, if we think about this for a moment, it should become apparent what number we need to put here.  If everyone tipped with the same percentage, then that means that each of the X values for all of our ordered pairs will be the same proportion of the corresponding Y coordinate, and that means that that line of best fit is going to go through each of our points exactly.  So we’re going to have perfect correlation.

Not only will we have perfect correlation, but the slope of our line is going to be positive, because the more the bill is, the higher the tip.  So we’re going to have perfect positive correlation.  What R value corresponds with perfect positive correlation?  This is going to be the highest value that R can be.  Since R by definition is bounded between -1 and 1 inclusive, the highest that R could be is 1.  And an R value of one represents perfect positive correlation.  So I'm going to submit that as my answer.  Fantastic!

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 use StatCrunch to perform hypothesis testing on standard deviations of alcohol treatments.  Here's our problem statement: Researchers conducted an experiment to test the effects of alcohol.  Errors were recorded in a test of visual and motor skills for treatment group of 21 people who drank ethanol and another group of 21 people given a placebo.  The errors for the treatment group have a standard deviation of 2.20, and the errors for the placebo group have a standard deviation of 0.78.  Use a 5% significance level to test the claim that the treatment group has errors that vary significantly more than the errors of the placebo group.  Assume that the two populations are normally distributed.

OK, the first part of this problem asks us to determine the null and alternative hypothesis.  The null hypothesis is by definition a statement of equality, so here it looks like we’re not going to select answer option C.  Of the options that remain, we need to look at the claim and compare it with the alternative hypothesis.  The claim that we see in the problem statement is that the treatment group has errors that vary significantly more than the errors of the placebo group.  Since the treatment group is mentioned first, we’re going to assume that that's going to have the subscript of 1 in the designations and the placebo group will have the subscript of two in its designations.  So 1 should be greater than 2.  And we see that's not answer option A.  It’s not answer option B.  It is answer option D.  Good job!

Now the next part of the problem asks us to identify the test statistic. To do this, I’m going to load up StatCrunch.  We don't have any data to put inside StatCrunch, but we are going to use the functionality of StatCrunch to get our test statistic.  To do that, I'm going to go to Stat –> Variance Stats –> Two Sample (because I’ve got two samples I’m comparing) –> With Summary (because I don't have any actual data).

Again, the treatment group was mentioned first, so we’re going to assume this is the Sample 1.  Notice we don't have the variance that's being asked for here in the options window, but we do have standard deviation, and we can get variance by squaring the standard deviation.  So I pull out my calculator here, and I’m going to take that standard deviation for the first group, and I’m going to square it.  That gives me the variance.  And the sample size is 21.  And then for the placebo group we see its standard deviation was 0.78, and I square it to get its variance.  And it also has a sample size of 21.

The hypothesis test — notice how it’s written a little differently than what we see here in our answer options from the previous part of the problem.  We can get the same thing, though, if we just take each of these expressions and divide by sigma sub 2 squared.  And so we’re going to leave this one alone.  But we need to change this inequality sign to “greater than” to match our alternative hypothesis.  And then I just hit Compute! And here in my results window the second to last number is always the test statistic. So I’m going to put that here in my answer field.  I’m asked to round to two decimal places.  Excellent!

The next part asks me to identify the P-value, which we can also get from our results table.  It's always the last number listed in the results table.  Here we see it's listed as “< 0.0001" — this means that the actual value of the P-value is not zero but a number that's so small it is for all practical purposes zero.  And so that's what I’m going to put here in my answer field.  Good job!

And now the last part of the problem asks us to make a conclusion about the hypothesis test.  Well, with a P-value of zero, we’re going to be less than whatever significance level we’re going to use for our test.  Here we’re having a 5% significance level, so zero is definitely less than 5% which means were inside the region of rejection, and therefore we’re going to reject the null hypothesis.  Whenever you reject the null hypothesis, you always “have sufficient evidence.”  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 hep, Today we’re going to learn how to use StatCrunch to perform hypothesis testing on two matched pair means of acting award ages.  Here's our problem statement: The following data list the ages of a random selection of actresses when they won an award in the category of Best Actress along with the ages of actors when they won in the category of Best Actor.  The ages are matched according to the year that the awards were presented. Complete Parts A and B below.

OK, the first part of Part A asks us to find the null and alternative hypothesis for a hypothesis test.  We’re asked to use a 5% significance level to test the claim that, for the population of ages of best actresses and best actors, the differences have a mean less than zero, indicating that the best actresses are generally younger than the best actors.  OK, so the null hypothesis by definition is a statement of equality.  And the claimed value here is going to be zero; we get that from the problem statement.  And then the claim is just going to be that the main difference is less than zero, which is the claimed value.  I check the answer.  Well done!

Now the next part of Part A asks me to identify the test statistic.  To do this, I’m going to take this data, and I’m going to dump it into StatCrunch.  Here’s my data in StatCrunch.  I’m going to resize this window so we can see everything a bit better.  OK, here's my data in StatCrunch.

So now we’re looking at matched pairs here.  And the reason why we know they’re matched pairs is that there's a unique value for the actor age for every value of the actress age, and they’re uniquely paired together.  So that tells me I have matched pairs.

So in StatCrunch I’m going to go to Stat –> T Stats –> Paired.  And here in my options window, I need to tell StatCrunch where my data is located.  So which goes in Sample 1, and which goes in Sample 2?  Well, generally it's going to be the order in which they are listed in the problem statement.  Here the actress is listed first, the actor is listed second, so that’s the order I’m going to put them in here in my options window.

We want the hypothesis test to make sure these values match what we have earlier.  And now they do.  I hit Compute! and here is my test statistic in the results window.  The second to last number in that table in the results window is always your test statistic.  I’m asked to round to two decimal places.  Nice work!

And now the next part in Part A asks me to identify the P-value.  I also get that from the table here in my results window.  It's the last value in that table that you see right here in the end.  I’m asked to round that number to three decimal places.  Well done!

And now the final part of Part A asks me to make a conclusion on the hypothesis test. We’re going to compare the P-value here with the significance level.  Our significance level, if you remember, here is 5%.  So we compare that with our P-value, which is about 1.6%., so definitely less than 5%.  So we’re less than our significance level.  That means we’re inside the region of rejection, and therefore we’re going to reject the null hypothesis.  Every time you reject the null hypothesis, “there is sufficient evidence.”  I check my answer.  Nice work!

Now Part B has two parts to it.  The first part of Part B asks me to construct a confidence interval based on the hypothesis test that we just conducted.  I could go through the menu options in StatCrunch again, or I could just go up here to Options –> Edit and I'm back into my options window.  I click the radio button for Confidence Interval, and now I need to make sure I have the right confidence level.  So back here in StatCrunch, our significance level is 5%, but we’re looking at matched pairs.

So normally we would just take the confidence level as the complement of our significance level.  But because we have matched pairs, we have to take the complement of two alpha — so I’m taking the complement of 10%, not the complement of 5%. The complement of 10% is 90%.  And I could put the extra zero here, but I don’t have to; it’s all the same number.  I hit Compute! and here in my results window the lower and upper limits of my confidence interval.  So I come back down here.  I put those numbers here in my answer fields.  Nice work!

And the final piece asks, “What feature of the confidence interval leads to the same conclusion reached in Part A?  Well, here we’re going to go back and look at our null hypothesis.  Remember, our null hypothesis is that the mean of the differences is zero.  So we look to see if zero is part of our confidence interval.  Zero is outside of the confidence interval; in fact, zero is up here ahead of our confidence interval.  So all the values that we get in our confidence interval are going to be less than zero.

So the confidence interval contains only negative numbers and, because zero's not a part of that, that means we’re, you know, 90% confident that zero is not going to be the mean of the differences.  Therefore, we could say that the null hypothesis is going to be rejected, because that’s what the null hypothesis says.  The null hypothesis says that it’s zero, but here zero’s not a  part of our confidence interval, so we’re pretty confident it's not zero.  And therefore were going to reject the null hypothesis.  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 use StatCrunch to perform hypothesis testing on two independent means of blood pressures.  Here’s our problem statement: Assume that the two samples are independent simple random samples selected from normally distributed populations.  Do not assume that the population standard deviations are equal.  Refer to the accompanying data set.  Use a 0.01 significance level to test the claim that women and men have the same mean diastolic blood pressure.

OK, the first part of this problem asks us for the null and alternative hypothesis, and it defines mu-1 as the mean diastolic blood pressure for the women and mu-2 as the mean diastolic blood pressure for the men.  Remember that the null hypothesis is always a statement of equality, so answer options A and C are not going to be correct.

Between answer options B and D, we need to look at the alternative hypothesis.  What is the claim that's being made?  Well, the claim that were testing here is that women and men have the same mean diastolic blood pressure.  So the claim is that the two are equal to each other.  But equality by definition belongs to the null hypothesis.  So therefore we have to take the compliment of the claim.  And so if the claim is that they’re equal, the complement of being equal is being not equal to.  So that means we’re going to select answer option B.  Good job!

Now the second part of the problem asks us to find the test statistic.  To do this, we’re going to take the data that they give us here, and we’re going to put the data into StatCrunch.  Here’s my window with StatCrunch, and the data is inside.  I'm going to resize this window so we can see a little bit better what's going on.

Alright, so now to get the test statistic, I need to perform the actual hypothesis test.  And the key question of course is “Do we know what the population standard deviation is?”  We’re asked to assume that they're not equal for each of the two groups, but other than that we don't know anything about the population standard deviation.  Therefore, we’re going to use t-scores to calculate our test statistic.

So in StatCrunch, I’m going to go to Stat –> T Stats –> Two Samples (because we have two samples that were comparing) –> With Data (because we have actual data).  Here in the options window, I need to select the columns where my data is located.  Which goes in Sample 1, and which goes in Sample 2?  Well, usually you follow the clue from the problem statement itself.  Here, the women are mentioned first, and the men are mentioned second.  So that’s the order we’re going to put them in here into StatCrunch.  And so the women go first, and then the men go second.

Make sure this box for “Pool variances” is not checked.  It used to be that the default was to check this box, and then they changed the coding in StatCrunch so that the default is now unchecked.  That’s what we want; we want that box unchecked.

Here under “Hypothesis Test,” make sure these values match what you see over here.  And notice the difference in the way that this is expressed.  So here we’re expressing it as a difference.  Here we’re expressing the two statements as being equal to each other.  They’re the same thing.  You just subtract mu-2 from both sides, and you’re going to get the same thing over here.  Not equal to, not equal to — that’s what we want.  I hit Compute! and here in my results window the test statistic is always the second to last number in this table in my results window.  I’m asked to round to two decimal places.  Excellent!

And now the third part of this problem asks us to find the P-value.  The P-value is back here in our results window.  It's the last value in the table in the results window.   I’m asked to round to three decimal places.  Fantastic!

And  now the last part of this problem asks me to make a conclusion about the null hypothesis and a final conclusion that addresses the original claim.  Well, to do this, we’re just going to take our P-value and compare it with our significance level.  We have a 1% significance level. Here we have a P-value of 35%, which is well above the significance level.  Therefore, we are outside the region of rejection, and therefore we’re going to fail to reject H-naught.  Whenever we fail to reject H-naught, there is not sufficient evidence (or there is insufficient evidence).  And I check my answer.  Well, that’s taking a long time.  Come on, come on, give it to me! Come on!  There we go!  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 use StatCrunch to perform hypothesis testing on means of female pulse rates.  Here’s our problem statement: Use the pulse rates and beats per minute of a random sample of adult females listed in the data set available below to test the claim that the mean is less than 77 bpm.  Use a 0.01 significance level.

OK, the first part of our problem asks us to provide the null and alternative hypotheses.  Remember, the null hypothesis by definition is a statement of equality, so obviously answer B is not going to be correct because this null hypothesis is not a statement of equality.  Of the remaining answer options, we look for the alternative hypothesis, and typically alternative hypothesis is to match the claim.  Here in the problem statement, the claim is that the mean pulse rate is less than 77 bpm, so were going to select the claim that represents the alternative hypothesis.  That’s going to be our correct answer.  So that's going to be this answer option here.  Well done!

Now, the next part of our problem asks us to determine the test statistic.  To do this, we’re going to take the data that they give us here and we’re going to dump it into StatCrunch.  So here's my data in StatCrunch.  And now I'm going to resize this window so we can see everything a little bit better.

OK, so now here in StatCrunch, I’m going to go up to Stat — and when you’re testing the mean, you could select Z Stats or T Stats.  To know which one to select, we need to ask ourselves the key question, which is “Do we know what the population standard deviation is?”  In this case, we don't know what it is; there’s nothing to indicate what it is in the problem statement.  And that means we need to use the Student-t distribution.

So I’m going to select T Stats –> One Sample (because I’ve only got the one sample) –> With Data (because I have actual data in StatCrunch).  Here in my options window, I'm going to select the column where my data is located.  And then I'm going to make sure — here under Hypothesis Test I’m going to make sure this matches what we've listed over here in the previous part of the problem.  That matches, then I hit Compute! and here comes the answers that I'm looking for.  So the test statistic is going to be the second to last number here in this table in the results window.  I’m asked to round to two decimal places.  Fantastic!

Now the next part asks me to determine the P-value.  The P-value is back here in the results window.  It's always the last value listed in the results table.  I’m asked to round to three decimal places.  Excellent!

And now the last part of the problem asks me to state the final conclusion that addresses the original claim.  To do this, I’m going to compare the P-value with the significance level.  Here in the problem statement, we have a 1% significance level, but our P-value was almost 26%.  That's definitely above 1%.  Therefore, we are outside the region of rejection, and we’re going to fail to reject the null hypothesis.  Because we fail to reject the null hypothesis, there is always insufficient or not sufficient evidence to support the claim.  The claim was that the mean pulse rate is less than 77 bpm.  I check my answer.  Fantastic!

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 use StatCrunch to perform hypothesis testing on proportions of jury selections.  Here's our problem statement: In a recent court case, it was found that, during a period of 11 years, 867 people were selected for grand jury duty, and 37% of them were from the same ethnicity.  Among the people eligible for grand jury duty, 80.7% were of this ethnicity.  Use a 0.01 significance level to test the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury.  Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that address the original claim.  Use the P-value method and the Normal distribution as an approximation to the binomial distribution.

OK, so this first part of our problem is asking us to identify the null and alternative hypothesis.  The null hypothesis, of course, by definition is a statement of equality.  So all these answer options where the null hypothesis is not a statement of equality (answer options B, for instance) will not be the correct answer.  So of the answer options where we do have the null hypothesis as a statement of equality, we need to look at the alternative hypothesis to select the correct answer.

Here we go back to the problem statement.  Typically, the alternative hypothesis reflects the claim.  And here we see the claim is that the selection process is biased against allowing this ethnicity to set on the grand jury.  If you’re biased against an ethnicity, that means you have fewer of that ethnicity than you should have.  So were going to look for the option that has the alternative hypothesis that says, “less than.”  And that’s going to be answer option A here.  Good job!

Now the second part asks us to find the test statistic.  And to do that, we’re going to look in StatCrunch.  So here I have StatCrunch open, and inside StatCrunch, to do my hypothesis test, I’m going to go to Stat –> Proportion Stats (because we’re dealing with proportions) –> One Sample (because we have only the one sample) and With Summary (because we don't have any data).

Here we need to supply the number of successes and the number of observations.  The number of successes is going to be the ethnicity that we find in the general populace.  And so that’s 37%.  But here StatCrunch is organized so we have to actually put in an actual number.  So I’m going to get my calculator out, and I'm going to calculate that — 37% of the total [is] 320.79, which rounds up to 321.  If the number had been such that I would round down, I’d still round up because I want to catch that partial member or person.

Of course, the total number we put here.  We want a hypothesis test, and here I need to make sure this area matches the answer option that I got from the previous part.  I hit Compute! and here's the results from my hypothesis test.  The z-score is the next-to-last number there listed in the table.  So I’ll stick that in here.  Fantastic!

The third part of the problem asks us for the P-value, which is right here, the last number on the edge of this table.  It says “less than .0001,” which is the same thing as zero for all practical purposes.  It’s technically not zero, but it's a number so low that it might as well just be zero.  So I can stick that here in my answer field.  Well done!

Now the next part asks, “What is the conclusion of the null hypothesis?”  Well, if I compare my P-value to my significance level — here in the problem we’re comparing with 1% — zero is definitely less than 1%.  Therefore, we’re in the region of rejection, and we’re going to reject the null hypothesis.  So I select that answer option here.  Well done!

And now the last part of the problem asks, “Does the jury selection system appear to be fair?”  Well, I go to the previous portion, right, where it says I’m going to reject the null hypothesis.  We reject the null hypothesis, so there is sufficient evidence to support the claim.  So I'm going to look to my answer options and select that option — “There is sufficient evidence to support the claim.”  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 use a z-score to complete hypothesis testing for a claim of inequality.  Here’s our problem statement: The test statistic of z = 1.25 is obtained when testing the claim that p does not equal 0.2978.  A: Identify the hypothesis test as being two-tailed, left tailed, or right tailed.  B: Find the P-value.  C: Using a significance level of alpha equals 10%, should we reject H-naught, or should we fail to reject H-naught?

OK, so Part A is asking for the type of hypothesis testing we’re conducting.  So to do that, we look at our claim.  And we see that the inequality for our claim is not-equal-to.  That’s the telltale sign that this is going to be a two-tailed hypothesis test.  If this had said that p is less than, then that would be like an arrow pointing to the left, and we would select the left tail test.  Or if it had said p is greater than, and that would be like an arrow pointing the right, and we would select right tailed.  But here we have not-equal-to, so that's the sign we have a two-tailed test.  Well done!

And now, Part B asks us for the P-value.  To get the P-value, I’m going to go back to StatCrunch and call up the Normal calculator.  You can use the z-score tables if you want — you’ve got the links here to that in your problem — but I find it's much easier to just use StatCrunch.  So here in StatCrunch, I go to Stat –> Calculators –> Normal.  And I want the standard Normal distribution, and that’s the default here in the calculator, so I don't need to adjust the mean and standard deviation.

But I do need to select this Between option here because we do have a two-tailed test.  So I’m going to select the Between option.  So now I’m going to get — I want the area for my tails.  The area that’s going to come out of the calculators is going to be the area between the tails, but that’s OK; we can just take the complement to get the area of the tails.  And that's were looking for.  This z-score is actually bounding the area of the tails, which is the P-value that we’re looking for.

So all I need to do here is stick in my z-score on the left, z-score on the right, hit Compute!, and this is the area in between the tails.  So to get the actual area of the tails, I’m going to take my calculator here and I'm going to subtract 1 from this number — or rather this number from 1.  And there's my area in the tails, which is my P-value.  So I’m going to put that here rounded to three decimal places.  Well done!

And now the last part, Part C, asks us to choose the correct conclusion below.  In the problem statement we’re asked to use a significance level of alpha equals 10%.  So we compare our P-value with the alpha level, and our P-value is greater than the alpha level.  So therefore we’re not in the critical region; we’re actually outside the critical region in between the tails.  So we’re not in the tails of distribution.  We’re actually in the center of our distribution, and so therefore we do not have sufficient evidence to support the claim.  We’re going to fail to reject the null hypothesis.  I check my answer.  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.