All right. Welcome back to our series on more traditional statistical inference. And in this video, we're going to talk about the normal distribution and how we can fit the normal distribution to our existing data. And for demonstration purposes, we're going to use a version of the card draw data set that we collected in the previous video on significance. All right, so we are here in Google Colab. We're using these libraries up here. So we've got numpy, pandas, plot9, and scipy.stats. And we're going to use this scipy.stats library to generate and fit normal distributions. The documentation can be found here, and also linked in the pages. And in particular, we're going to use the stats.norm.pdf to fit a normal distribution to the data. So I copied down some code from the significance level test where we drew six red cards in a row. And I am just going to multiply it by 10, so that we have a. sufficiently large data set. You've got 60 data points then. And then we conducted a binomial test, calculated the proportion, and converted that into a data frame. And so, to fit a normal distribution, we need to know two values. We need to know the mean and the standard error. These are the two main parameters that control the shape of a normal distribution. So we always need to find them. So we can say the mean r for red is just phatdf p hat dot mean. And then likewise, the SER for red is phatdf phat dot std. I made a typo right there. And then to show these, we can print the mean, it's just mean r. We can print the standard error and go ahead and run that code. So we can see that the average phat was 0.5, and that the standard error is 0.06. And these are proportions, so this makes sense.

When we run our binomial distribution, we expect it to be centered around whatever our null hypothesis assumption is, which in this case was 0.5. So, before we can then visualize this normal distribution, we need to create new data that fits the normal distribution to our existing data. And so, we say phat data frame, a new column called x underscore pdf. This is the x variables of our normal distribution. And so, to create them they're just evenly spaced variables, so we say np numpy.lin space. And in this case you give it the first value, the second value, and how many data points you want. So, in other words, this will give us 1000 evenly spaced data points between 0 and 1. And then we also need a y variable, so y pdf and this needs to be p hat df. And this is where we actually fit the normal distribution. We say, stats dot norm dot pdf. And then we give it the x values that we already created, x underscore pdf. We give it the location, which is the mean value, and we give it the scale, which is the standard error value. So then those run as expected. We can now look at what p hat looks like. If we just, you know, print the first five rows we can see that we've got our proportions here, and then we've got our x values that run from zero to one thousand, or zero to one with a thousand in between, and then we've got the pdf values that are associated with these x values but follow the same shape as p hat.

So then we can go ahead and visualize this. And so, we'll do ggplot with our p hat data frame. And the first thing that we're going to plot is a dotplot so we'll say, genome dotplot aes, x equals p hat and dot size is 0.25. So I'll go ahead and run this to give you a look about what this looks like. So we've got a fairly normal-esque distribution of our data points. So to add the actual normal distribution onto it, we can add a line plot, say geomline aes. And our x value is just that x pdf. And our y value is the y pdf. And then we can say color equals, red. And I'm going to add a size equal to one, make it show up a little better. So here that should be x. And so now we can see how this normal distribution looks with our actual data. So we've got our actual data in these black dots, and then we've got this normal distribution. And what has been done with this, y stats norm pdf, which created ypdf, is we specified the center of the normal distribution to be centered on the mean value of our real data. And the standard error, the spread of this normal distribution, to be equal to our sampling distribution. And therefore, we can see what this idealized normal distribution would look like. And therefore, can look at where we're maybe over estimating normal, where it's less normal, and where it's possibly more normally distributed.

All right, we're going to continue to talk about the normal distribution in this video. And in particular, we're going to learn how to actually calculate the confidence interval from the normal distribution. And so, in the past we have defined our 95 percent confidence interval as the mean, plus or minus the two times the standard deviation. And so, in effect that looked something like using our data from above where we calculated the mean and standard error of our p hat distribution, meanR minus 2 times SER comma meanR plus 2 times SER. Then we can print the confidence interval. And so, here we can see that the 95 percent confidence interval is 0.37 to 0.63. But this is an approximation. Normally our value here, is usually calculated from something known as a z star, z, or a z statistic. So in order to do that, we need to specify the interval, P equals 95. And then we calculate the zstar which is just using stats dot norm dot interval of P divided by 100. And what we want to do is convert this into an array so we can run that. We can look at zstar which is minus 1.95 plus 1.95. And we can see how that is very close to 2. So what we have been doing so far in this class is using two as a very easy approximation for the actual z-star value, which is 1.95996398. So if we want to do a confidence interval with this zstar, we can really use the same formula. I'm going to copy this down. But instead of 2, we can use the actual value. So, and we don't need to do an array because this value already has the negative and the positive here. So I'm going to erase that, still say meanR but replace 2 with zstar times the standard error for R. And so, we can run this. And we can see that it has flipped it until we do this, change this to a plus. It goes back to the order, smaller versus larger because that's adding a negative and then adding a positive. And so we can see that it actually is very similar once it gets out to this third digit is where we start to see differences in the confidence interval. But the benefit to doing this type of methodology is that we can easily change our confidence interval to whatever confidence level we want by just changing this p-value. So if we want the 80th confidence interval, all I need to do is change that to 80. My zstar changes and now my confidence interval has changed. And so, this is a very easy way to do confidence intervals. And it's something that you would normally do in a traditional statistics class where you would look up this zstar value or your dip in confidence level and do a calculation. But it's important to note that this style of doing confidence intervals assumes that your data is normally distributed, and that it is sufficiently large enough to do this type of statistical analysis on it because it is following the central limit theorem, which we talked about earlier in the lesson.

Hi. In this video we're going to cover merging data frames. So, we're going to continue with our state energy ranking data from the Energy Information Administration. In this video, we're going to merge the natural gas data frame ngdf, with the total energy data frame tedf. Let's exemplify this first with the inner merge. This is the default method. And we'll do this by creating a new data frame, which is called DF, generically PD. To reference pandas, merge these two objects: ngdf, tedf. We want to merge the natural gas with the total energy. We also need to specify on which variable to merge. So, let's do this by state, that is, it's going to match Texas and natural gas with Texas, and total energy Pennsylvania in the first, Pennsylvania the second, and so on, and so forth. So it's going to pair them off on a basis of state. Let's see what this looks like. Here we go. So we've got first the rankings. We've got rank underscore X. That's interesting. What does that mean? Well, these are the rankings from the natural gas data frame. That's our first data frame. So it gives it the subscript X as opposed to rank Y. So this is the rank from the second one. Note that these two variables were called the same thing. So it automatically appended these X and the Y to distinguish the two. Then we have state. This was our variable, our key variable, that we're performing the merge on. So we have all the states here, and then we've got our natural gas and our total energy. And now this miraculous function has automatically paired the values according to what state they belong to.

However, there's a problem with this. We scroll down, we don't have all the states. This stops at 33. Note that the indexes for this start at zero. So we've got 33 states here. There really ought to be 51, 50 states, plus DC. So what happened? Why'd you get rid of those? Well it only kept the states that were equal in natural gas and total energy. Recall that in the natural gas data frame we had some states that had their full names in there. Let's try this a different way. Let's try it with the outer join. So say it's DF again PDF for pandas dot merge. And again, we'll have our ngdf, or tedf, on equals state. And now we'll say, how is outer. So we're going to do an outer merge. Let's see what result we get here. So we still have our rank X, our rank y, that all looks the same. But now we've got down to 69 states. Well, that's strange. That's too many. I'll click on this thing to just get a better inspection of the output here. What's going on, if we scroll down, we need to display some more values here. We see that we've got these states by name here, that don't have any data, no natural gas production, no total energy production. And then we've got their corresponding acronyms. So how did this happen? Well, the natural gas production data set had these states listed out in their full name because they didn't produce any natural gas, so it listed them differently for whatever reason. Whereas the total energy data set does have data for these states and uses their two-letter abbreviation. And the merge function can't associate these two things because it's just not that intelligent. It doesn't recognize that Hawaii is abbreviated HI. We would need to tell it that. So, what do we do? Well, this will be for the next video, logic and subsetting. We'll solve that problem there. For now, that's it for merging.

Single Proportion

• Sample Statistic: $$\hat{p}$$
• Standard Error: $$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
• Confidence Interval:$$\hat{p}\pm z^{\ast }\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
• Hypothesis Test Statistic:$$\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

Single Mean

• Sample Statistic:$$\overline{x}$$
• Standard Error:$$\frac{s}{\sqrt{n}}$$
• Confidence Interval:$$\overline{x}\pm t^{\ast }\times \frac{s}{\sqrt{n}}$$
• Hypothesis Test Statistic:$$\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$$

Difference in Proportions

• Sample Statistic:$${\hat{p}}_1-{\hat{p}}_2$$
• Standard Error:$$\sqrt{\frac{{\hat{p}}_1(1-{\hat{p}}_1)}{n_1}+\frac{{\hat{p}}_2(1-{\hat{p}}_2)}{n_2}}$$
• Confidence Interval:$$({\hat{p}}_1-{\hat{p}}_2)\pm z^{\ast }\times SE$$
• Hypothesis Test Statistic:$$\frac{({\hat{p}}_1-{\hat{p}}_2)-0}{\sqrt{\frac{{\hat{p}}_1(1-{\hat{p}}_1)}{n_1}+\frac{{\hat{p}}_2(1-{\hat{p}}_2)}{n_2}}}$$

Difference in Means

• Sample Statistic:$${\overline{x}}_1-{\overline{x}}_2$$
• Standard Error:$$\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$$
• Confidence Interval:$$({\overline{x}}_1-{\overline{x}}_2)\pm t^{\ast }\times SE$$
• Hypothesis Test Statistic:$$\frac{({\overline{x}}_1-{\overline{x}}_2)-0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$$

Paired Difference in Means

• Sample Statistic:$${\overline{x}}_d$$
• Standard Error:$$\frac{s_d}{\sqrt{n_d}}$$
• Confidence Interval:$${\overline{x}}_d\pm t^{\ast }\times \frac{s_d}{\sqrt{n_d}}$$
• Hypothesis Test Statistic:$$\frac{{\overline{x}}_d-0}{s_d/\sqrt{n_d}}$$

Hello, and welcome back to the lecture series. In this lesson we're going to talk about more of the traditional statistical side of hypothesis testing. So normally if you look at a statistics textbook it won't teach you how to conduct these randomization procedures. Instead, they'll focus on the statistical theory. So the first part of this lecture is actually having, we'll focus on significance.

So we're here in Google Colab. And so, in order to focus on significance we're actually going to look at a deck of cards. So we're going to use a proportion. So we've got our proportion, is the number of cards, number of red cards divided by the total number of cards. And in theory, it should be 50-50. So a standard deck of cards has half-red, half-black colors. And then our alternative is that red's more than 50 percent. In effect, that we've got a deck that isn't fair. So, we're going to go ahead and get started with that. So I'm going to type in x equals np dot array. And this is how we're going to create an array before we get started. So, I have a deck of cards here. So I'll shuffle it. And I'm just going to draw the top card. And we're going to record whether or not it's red or black. So, first card is a five of diamonds. That is red. Second card, six of diamonds, also red. Six of hearts. Nine of diamonds Need to have a comma there. Nine of hearts. And we'll do one more, four of diamonds.

So we'll go ahead and stop there. And we're going to use this data to look into what it means, how significance levels of how significance testing can change based off of our data that we have here. So with that, we can go ahead and get started. All right, so we have this data here that I threw out of my deck of cards. And you know the first six draws we had were red data points. And so we can go ahead and develop our hypothesis test. This is a single proportion test so we need to know the sample size which is just the length of our x-array. We need to know how many iterations we're going to do. So that's capital N and then we need to integrate or conduct the binomial test where we specify our sample size, our null proportions, and the number of iterations. So we can run, this looks like it's still connecting, and I need to run my libraries here.

So we've got numpy pandas plot nine. And later in this lecture series we'll get into scipy dot stats. So run this, so this sets up our binomial data set and then we need to actually calculate our proportions. So we can say p-hat is just counts divided by the sample size. And our data p-hat is the actual data set. So we have the length of x in which x equals r, and then we divide that by little n. So, we've got our data here, and as we have done in previous lectures we can go ahead and visualize this. So we need to create a data frame. So we say pd dot data frame of just our sampling distribution of proportions, and I'll go ahead and rename these columns to just say p hat. And then we can do our gg plot. And so we're going to plot our p hat data frame and we're going to do a dotplot. So geom underscore dotplot. We have our AES where we just need to provide the x value, which is p hat. And we can set the dot size to something relatively small. And then we can add our vertical line where the x-intercept is set to data p hat, the color is set to blue, and the line type is set to dashed. So we can run that. And we can look at this, and we can already see that you know the majority of our data is certainly below our sample statistic where 100 percent of our data came from the red draws. So we can already sort of guess that our p-value is going to be quite low. But let's see how low it actually is. A

nd so, we can say print the p-value. And this is just the length of p hat data frame where the p hat column is greater than or equal to our sample proportion, divided by the number of iterations that we did. And so we can see this p-value is .021. Where we therefore reject the null hypothesis in favor of the alternative that the proportion of reds is actually higher than 50 percent. Well, let's say for example, that I only did this four times or three times. So we just drew three cards. We can go ahead and run this, and rerun all of these different things. We can see how our proportions are sort of spreading out. They're not becoming as normally distributed as we would expect. And then we run this, and suddenly our p-value is greater than 0.5. Which means we fail to reject the null hypothesis that there's more than 50 percent of the deck. So if I go back here, and you know, let's say we continue to draw cards. I just drew a ten of diamonds an eight of hearts, and then a queen of hearts. So now we have even more reds. We're up to nine reds drawn. Run through all of this. We can see it's becoming a little bit more normally distributed in the plot. And now our p-value is zero. One hundred percent of the data is below there. So we've got as low as we can go, which provides really strong evidence that this deck that I'm drawing from is not actually a fair deck. There are much more reds than we would get in a normal randomly distributed deck. And if we continue to increase the number of draws, if they continue to be red, we'll continue to have this p-value drop to zero.

And so, this is really good to evidence how you know the significance of our data set of our hypothesis test can change depending on how large your data set is, but also, how where you set your significance level. And so, if we go back to when we had six red draws and run through these our p-value is .016. Which under a normal significance level we would reject the null hypothesis. But sometimes people will use 99 significance levels which becomes 0.01. And in this case, if we followed that significance level we would fail to reject the null hypothesis, just barely. And so, this is why, you know, determining that significance level is really important. And sticking to it, and acknowledging what your, which level you're working at. But also making sure that you get a large enough sample size so that you feel confident in your final statistical conclusion.

All right, in this series of videos we're going to shift back into hypothesis testing. And in particular, we're going to focus on the one-line tests that you can do to do all of the hypothesis testing that we learned in the previous lesson, but in just a single line. And while a lot of times we'll still want to do the randomization procedures, these one-line tests can be a good way to to check your answer. Because although they may be slightly different, a lot of times they should actually follow the same pattern as your randomization distribution.

So, let's go ahead and jump right in. So, we're here in the code for hypothesis testing. And in particular, I'm going to focus on the hypothesis test for a single mean in this video. And so, when we write our hypotheses, we say that mu equals in this case 6.1, the capacity. And then our alternative is that mu is less than that. So, in essence, we're going to test whether or not wind actually met the capacity in, during the Texas 2021 cold snap. And so, in the previous lesson we went through these steps: we shifted the data, we initialized the variable, and we implemented this randomized random choice function where we set replace equal to true and focused on the shifted data. And then, eventually, we calculated the p-value as the proportion of data that was less than our original mean. And we got a p-value of zero, which leads us to reject the null hypothesis in favor of the alternative, that the capacity of wind, that the average wind was less than the capacity.

And so, this one-liner test, we're going to use a library, the stats library in Python. And in particular, this library is the stat SCI Pi stats, which is the same library that we used when we were writing our normal or working with the normal distributions in the previous videos. And so, in order to do the single mean hypothesis test, I'm going to put the results into a variable called results. And we say stats dot ttest underscore one - the number one - samp. So effectively, a one sample ttest. Then we give it our data, which is wind - which is the observed wind generation. We give it our population mean, which is what we expect. So this is our capacity in this case, or whatever value you have set in the null hypothesis. And then we give it the alternative. And in this case, we specify the alternative as less. Meaning that we are doing a test in which we're looking at less than, we're using a left-tailed test. And then to actually see the p-value we can just say results dot p-value at the end. And so, we can run this. And we can see that the p-value is still less than our significance level. It's very, very close to zero. But as what often happens in these one-liner tests, that is more specific. So, instead of just staying zero, we've got you know quite a few digits, but still very, very, very close to zero given the e to the negative seven. And this is a very common result when we're using these one-liners and comparing them to randomization procedures, because what this one-liner test is actually doing is conducting that same randomization procedure that we went over, but with a lot more data. And so, essentially, it does something very similar to this. And instead of being a thousand, is maybe a million, 10 million. It essentially generates a lot more iterations, which allow it to become a lot more specific in the results.

We're back in our one-line test, hypothesis test video. In this video, we're going to focus on the hypothesis test for a single proportion. And so, here our null hypothesis is that the proportion is equal to 50 percent, and our alternative is that it's less than that, in effect, looking at how frequently windfall wind generation fell below capacities during the Texas 2021 cold snap. And so, in our previous lesson we went over how to do the randomization procedure for this test. And we start off with simulating random data using random binomial. Then we calculate the original, or the sampling distribution of proportions, calculate the data sample, and then we plotted it. And eventually got a p-value equal to zero, which led us to reject the null hypothesis that the wind actually fell below capacity less than 50 percent of the time.

So, if we want to make the one-liner test for this, it involves a couple extra steps. We're still going to be using that stats.cypi library, but before we can actually implement the test we need to define two variables, the first is success rate. And so, this is how often we were successful. And so here, when success rate is defined by the length of wind in which wind was above 6.1, above capacity, and then we need to define the sample size, which is just the total number of variables or data points. And so then our actual one-line test is stats dot binome underscore test, so binomial test. We give it x equals our success rate, n equals our sample size, and again we specify the alternative equal to less. And if you were doing a right-tailed test you would change this to greater, for example.

But in this case, following what we did up here, we're going to do a left-tailed test. So we can run this. We've got a warning that this command is currently being phased out. So eventually we'll need to use this new command, binom test, but for now it's still working. And essentially, it automatically prints out this p-value here. And so, because of that we can compare to this and look that they're very similar. Once again our one-line test is showing a bit more specificity in that number, but is still very, very close to zero. So again, we reject the null hypothesis.

In this video, we're going to go over the one-line test that can be used for a two sample mean hypothesis, a comparison of the means. And so, when we wrote these null hypotheses, our null hypothesis was that the mean of gas minus the mean of wind, was equal to zero. Thus, there was no difference between the two data. And our alternative is that gas minus wind is less than zero, suggesting that there is a difference between the two.

And here we calculated mean for the percent deficit as generation minus capacity, divided by capacity. And so, we did all of this prep work, and then we got into the actual randomization procedure where we calculated our sample difference, did our reallocation procedure where we sampled from percent deficit with replace equals to false, and recalculated these differences, and ultimately got a p-value of zero. Which suggested that we reject the null hypothesis. In terms of this one-liner test, I'm going to call the results, results. And again, using that stat dot scipy library we can say stats dot t-test. And in this case, we say IND, for independent samples. And then we give it our first sample, which is sample gas. And a key point here is to make sure that you're matching the order that you write your alternative hypothesis in. And so, we did gas minus wind. So our first sample is gas. And I want to also point out that we are using the actual data here. So our sample is located from the samp.loc which we calculated in step one. We're not using any of the reallocation samples which we calculate in step three. So our first sample is samplegas, our second sample, samplewind. Here we need to say equal there equals false. And this just tells it that the variance is not equal between the two data frames to variables. So, it'll conduct a slightly different version of the t-test than if they were equal variance. And then we specify the alternative equals less, since we are using a left-tailed test. And then after that, we can say results dot p-value to print the p-value.

And so, here we can see that we are very, very close to zero, following that same pattern that we have seen in the earlier two tests. It still has the same conclusion as our randomization procedure, but with more specificity. So again, we reject the null hypothesis.

In this video, we're going to talk about the one-line test for doing a mean of differences, or a paired comparison test. And so, in this test we're still using those natural gas and wind data we needed to calculate the difference, and then take the mean of that difference. And then we did this randomization procedure where we calculated a multiplier to figure out if that difference was equal to zero or not equal to zero. And eventually we calculated a p-value, which resulted in rejecting the null hypothesis. And so, this one minor test, again as I've done with most of these I'm going to call it results, and say stats dot t-test. And in this case, it's rel to signify that this data is not independent that we are doing a paired comparison test we still give it our first sample, sample gas, making sure that it follows that order that we've talked about before. our second sample, sample wind. And in this case all we need to do specify the alternative equal to less and do results dot key value. And so, we can see what our p-value is. It's very close to zero. We reject the null hypothesis, but we can note that there is a little bit more specificity with this one-line test than with our randomization procedure.

In this last one-line hypothesis test, we're actually going to get in to a difference of proportions. And so, here our null hypothesis is that there's no difference between the proportion of gas that was below capacity, and the proportion of wind that was below capacity. And our alternative is that it was greater than zero. And so, here before we can actually get in to the one-line test, we do need to do some prep work. And so, I'm going to create a new column called, below capacity for final_gen. And this is essentially just going to be equal to the deficit, but we're going to use a special function called an apply function to quickly apply a conditional statement to the deficit. So we say dot apply, and then we give it this keyword, lambda. Which effectively tells python that we're about to create an inline function. And we say x colon true, if x is less than zero, else false. And so, what this is doing is, it's, x is just a placeholder, so it just needs to be the same across both values. And it's saying set x, set below capacity to true, if x, x being deficit, is less than zero. Otherwise, if x is greater than, or equal to zero, set below capacity equal to false. And so, if we print out the first five rows of our data set, we can see that we have our date time column, we have our fuel generation capacity deficit, percent deficit that we've seen before, and now we have a below capacity, which will be true anytime the deficit is negative, and false any time the deficit is positive.

So then once we have our data as it needs to be, we need to actually calculate all of our samples. Similar to our difference of means, we needed to calculate the sample mean. Now we need to calculate the sample proportions. And so, in this case we can see sample wind prop is just final gen.loc and it's where final gen dot fuel, equals wind below capacity. So we're only interested in the below capacity column. And so, we're just going to leave it at that. So we're essentially going to just extract a below capacity column for wind. I'm going to copy this, paste it, and change the instance of wind to gas. And here's wind to gas. So that we just pull out the natural gas data. And then we're actually going to calculate the proportion, so our p wind of our sample is the length of which sample wind prop. And, we are only interested in where sample wind prop equals true. And then we divide that by the total length of sample wind prop. And I'm going to copy that, and change this to gas here, gas here, gas, and finally gas. So these will give us the actual proportion of when the wind in natural gas was below capacity. And then I'm going to get diff prop which is following our alternative hypothesis up here, is gas minus wind. So it's p gas samp minus p wind samp.

So we can run all of that and see that the difference in proportion was 0.35. So then we can move on to the one-line test. So we won't actually do a randomization procedure for the proportion. But in theory, you could follow the procedure for a difference of means, and just substitute in your difference of proportions. So in order to do this one-line test, we need a new library. So we're going to import stats models.stats.proportion. And we're going to call it prop. So this is a new library that is specifically used for a difference of proportions. And in order to use the test that we're going to do, we need to define our success number. Which is very similar to what we did for a single proportion where we determined our success rate. In this case we need to determine the number of successes for our first data set, samp gas, and the number of successes for our second data set samp wind. And so, we're going to do that within square brackets. So again, the key here is to follow the order that you went in for your hypotheses. So we want sample the length of sample gas prop, where sample gas prop equals true. So this is the exact same thing that we did up here to calculate this p wind, or p gas. And then I'm going to copy this, everything from the link command comma, and then paste. And I'm going to edit this, so that we have instances of gas equal to change those two instances of wind. So these are the number of successes for both gas and wind. And then we also need to define sample size, which is just the length of sample gas prop, and then the length of sample wind prop. And so, this is, in effect, the denominator of our proportion up here. And so, once we have these values, we can actually implement the command, which from the prop library that we defined up here. So it's prop dot proportions underscore z-test. This is the command. We need to give it a variable called count, which is our success number, and we need to give it a variable called n ops which is our sample size. And then we need to give it the alternative, and in this case, we follow a slightly different form of alternative. We're using a right-tailed test so it's a greater than sign. But for this particular command we need to say larger. And so, instead of saying greater or less, we say larger. And in particular, we want the first value that's going to print to results. And so here we can see this p-value is very close to zero, much less than our significance level. So we can reject the null hypothesis in favor that the proportion the difference in proportions is greater than zero.