[DATA] Concrete As concrete cures, it gains strength. The foll...

Question

[DATA] Concrete As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

Table showing paired 7-day and 28-day concrete strength values in psi for testing linear relationship assumptions.

d. Assuming the residuals are normally distributed, test whether a linear relation exists between 7-day strength and 28-day strength at the alpha = 0.05 level of significance

Accepted Answer

Hello. In this video, we are told that an engineer records the compressive strength of a batch of bricks at 2 days and at 10 days. The results for the 6 bricks are labeled in the following table. We want to test at the significance level of 0.05, whether a linear relationship exists between the two day and the 10 day strengths, assuming normality of residuals. Now, in order to determine if there is significant evidence, whether there is a linear relationship or not, we need to come up with a hypothesis test on the population regression slope, which is going to be beta one. So let's go ahead and start by making some hypothesis. For the null hypothesis we are claiming that B1 is going to equal to zero if there is no linear relationship. And what that means is that for our alternate hypothesis, B1 will not equal to zero if there is significant evidence that there is a linear relationship. All right. Now, in order to test these claims, let's go ahead and take a look at the given information. Now, we are told that we have a significance level of 0.05. And if we take a look at the table, both the 2 day and the 10 day strings have a sample size of 6. So our sample size N is going to equal to 6. We can use this amount to solve for the degree of freedom to help us find our test statistics as well. In this case here, because we have two different batches or 2 different days that we're testing, our degree of freedom is going to be defined as N minus 2. That is going to be 6 minus 2, which is going to give us a degree of freedom of 4. All right. So what we need is we need to go ahead and first define our test statistic for testing this hypothesis. Now, our test statistic is going to be defined as T equal to B1, divided by the standard error of the slope, which is going to be S sub B1. Now, here, B1 is going to be defined as the following. It is the summation. Of our elements of X minus their mean multiplied by the elements of Y minus their mean. And this will be divided by the elements of X minus their mean quantity squared. So, what we need to do is we need to calculate B1, but what about the denominator as sub B1? Well, here we need to find a chain of things, but as a be one. Is going to be defined as the square root. Of our mean squared error, which is S E squared divided by the sum of the elements of X minus their means squared. Now, here, our mean squared error is going to be defined. As The sum of squared errors divided by the sample size minus 2, so divided by a degree of freedom. And or sum of square errors is also defined. As the sum Of the elements of Y minus their mean quantity squared. Minus B1 multiplied by the sum of the product. Of the elements of X minus their mean multiplied by Y, the elements of Y minus their mean. So there are a few summations that we need. We're going to need the product of the sums for B1. And the the squared sums of X for the denominator of B1. We are also going to need the sum. Of the elements of y squared and once we have all the summations, we can calculate B1 which we need for our sum of squared errors which we then need for our mean squared error which we finally need for the standard error of the slope in order to find our test statistic. So, let's go ahead and find all these summations first, but first, before we find the summations, we're going to need to calculate the means for X and Y. Now, in order to calculate the means, for X, we need to take the sum of all of the values of the two day strengths, which is going to be a total. Of 50,030 divided by 6. Which will then approximate to 838.33. Then we will take the mean of the 10 day strings, which we will call Y bar, and that is going to be a total of 7690 divided by 6, which will then approximate to 1,281.67. Now that we have these means, we need to go ahead and calculate all of the summations. So let's go ahead and organize this data onto a table. So what I provided here is all the necessary summations that we need to get started. Now, let's go ahead and use this table to calculate B1. So B1 is going to be the product of the sums. Of X and Y and that is going to be 7000. 16.6666 divided by the sum of the X elements squared, which is going to be 3,483.3334. And by using a calculator, B1 is going to have an approximate value of 2.0144. Now that we've calculated B1, we have the numerator for our test statistic, and we needed necessarily value for SSE. Now, let's go ahead and calculate the sum standard error. SSE is going to be the sum of the Y elements squared, which is going to be 1,428, 14,283.3334. Minus the value of B1, which is 2.0144, multiplied by the product of the sums of X and Y, which is going to be 7,016.6666, and this is going to give us an approximate sum of standard errors of 148.9602. Now that we have that, we can calculate the sum of the errors squared. So S E squared is going to equal to 148.9602, and this is going to be divided by a degree of freedom, which is 4. And by using a calculator, this is going to approximate to be 37.24. And finally, now that we have the sum of the error squared, we can go ahead and calculate the standard error of the slope. So the standard error of the slope. Is going to equal to the square root. Of 37.24. Divided by the sum of the elements of X squared, which is going to be 3,483.334, and with the help of a calculator, this is going to approximate to be 0.1034. Now that we have this, we can finally calculate our test statistic, which is going to be T is equal to B1, which is 2.0144, divided by the standard error squared, which is 0.1034. And this is going to give us a test statistic of approximately 19.4816. So now that we have this test statistic, well how do we determine whether there's significant evidence Of a linear relationship or not. Well, I'm gonna go ahead and first clear up some room because I am running out of space here, so I will be deleting this table now. But what we need to do next is we need to go ahead and calculate. A critical value for a two-tail test. So we need to find our critical value now in order to compare our test statistic. The critical value, we need to find a critical value for a two-tail test with a significance level of 0.05 and a degree of freedom equal to 4. And by using a T table, that is going to give us a critical value of positive and negative, 2.776. Now, here, if the absolute value of the test statistic. is greater than the than the test statistic itself, the T critical value. Then we rejected no hypothesis. So here, Our T calculated is 19.4816, and that is greater than the T critical value we calculated, which is 2.776. And so with that being said, we go ahead and we reject the null hypothesis, and we see that there is significant evidence to suggest that there is a linear relationship between the two and 10 day compressive strengths. And so with that being said, the correct solution to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Linear Regression and Correlation

Hypothesis Testing for Regression Slope

Assumption of Normally Distributed Residuals

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