The null hypothesis for an anova is that all treatments/samples come from populations with the same mean. the alternative hypothesis is best stated as: "" of the population means is different from the others.

Alternative hypothesis: "AT LEAST ONE" of the population means is different from the others

Step-by-step explanation:

Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".  

The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"  

If we assume that we have n groups and we want to check if the population means are equal, th best way to check this it's with an ANOVA test.

The hypothesis for this case are:

Null hypothesis:

Or in words:

Null hypothesis: All treatments/samples come from populations with the same mean

Alternative hypothesis: Not all the means are equal

Or we can say:

Alternative hypothesis: "AT LEAST ONE" of the population means is different from the others

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1