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# How To Calculate P Value For Linear Regression | Using A P-value To Make Conclusions In A Test About Slope | Ap Statistics | Khan Academy

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## Using A P-value To Make Conclusions In A Test About Slope | Ap Statistics | Khan Academy

Transcript:

“Alicia took a random sample of cell phones and found a positive linear relationship between the speeds of their processors and their prices. This is computer output from the regression analysis of its sample least squares method. ” Let’s find out what’s going on – she took a sample of phones, they don’t tell us exactly how much, but she picked up a number of phones and found a linear relationship between CPU speed and price. This is the price, this is the CPU speed. And then presented the data from his sample graphically. Each phone will have one point of information and then entered these points information into the computer and he was able to derive regression rights for her sample. And the regression rights for her sample, if we say that this will be y, or y with a hat, and it will be a plus bx, for her sample a will be 127,083. That’s it, that’s it. And for her sample, the angular coefficient of her regression rights will be the speed factor. Another way to look at this – this variable x here is the speed, the coefficient of this is the angular coefficient. But we must remember that these are approximate calculations of some probable truth in the universe. If she could sample any phone on the market, then it would receive the real parameters of the general population, but since this is a sample, this is just an estimate. And since she sees this positive linear relationship in her sample, this does not really mean that it is a naka for the whole general population. It may just have happened that things in the sample to have had such a positive linear relationship. And so she tests a hypothesis. In a hypothesis test, you accept that there is no dependence between CPU speed and price. The beta here will be the real indicator of the general population for the regression of the general population. If this is the general population, where the price is on the vertical axis, and the processor speed is on the horizontal axis, and if you can look at the whole population – I don’t know how many phones there are, but there could be billions of phones – and then you make regression rights, then the null hypothesis is that the angular coefficient the regression rights will be 0. Regression rights may look like this, such as the equation of regression rights for the general population, y with a hat, it will be alpha plus beta on x. And the null hypothesis is that the beta is equal to 0, and the alternative hypothesis, which is her suspicion, is, that the angular coefficient of the regression rights is in fact greater than 0. “Assume that all conditions for inference have been met. Alpha is equal to 0.01 level of significance is there enough evidence to conclude that there is a positive linear relationship between these variables for all mobile phones? Why? ” Stop the video and see if you can solve this. To do this task, we must say, “Assuming that the null hypothesis is true, assuming that this is the real angular coefficient of the regression rights of the general population “- I guess you can think about that – “What is the probability of getting this result here?” And we can use that information and our approximate calculation of the angular coefficient of the regression rights of the sample distribution of the sample, and we can find the t-value. And for this situation, where our alternative hypothesis is, that the real angular coefficient of regression of the general population is greater than 0, we can look at the p-value as the probability of getting a t-value, greater than or equal to this. That is, to obtain a t-value, greater than or equal to 2,999. You may want to say that there is a column that gives us a p-value, and you may have found for us, that the probability here is 0.004. But you have to be very, very careful, because here they actually give us – I guess you can say that – bilateral p-value. If you think of a t-distribution – and they would do it for the appropriate degrees of freedom – this says what the probability is that we will get a result, in which the absolute value is 2,999 or higher. If that’s t = 0 here in the middle, and that’s 2,999, we are interested in this area, we are interested in this right tail. This p-value gives us not only the right tail, but it also tells us what is the probability of getting something less than -2,999, or including -2,999. This gives us both of these areas, so if you want the p-value for this scenario, then we will look at this. And, as you can see, because this distribution is symmetrical, the t-distribution will be symmetrical, you take half of that. This will be equal to 0.002. And in each significance test you will then compare the p-value with its level of significance. And if you look at 0.002 and compare it to 0.01, which of these is bigger? First you can say that 2 is greater than 1, but these are two thousandth against one hundredth. That’s 10,000 here. That is, in this situation our p-value is less than the level of significance and we say: “The probability of getting such or a more extreme result is so low if we accept the null hypothesis, that in this situation we will decide to reject the null hypothesis, which will suggest the alternative. ” “Is there enough evidence to conclude that there is a positive linear relationship between these variables for all mobile phones? ” Yes. “Why?” Because the p-value is lower than our significance level and we will reject the null hypothesis, which suggests the alternative.

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