Find the t values that form the boundaries of the critical region for a two-tailed test with a=.05 for n=6 n=12

n=6

"=T.INV(1-0.025,5)", and we got the critical values given by:  

n=12

"=T.INV(1-0.025,11)", and we got the critical values given by:  

Step-by-step explanation:

Previous concepts

The t distribution (Student’s t-distribution) is a "probability distribution that is used to estimate population parameters when the sample size is small (n<30) or when the population variance is unknown".

The shape of the t distribution is determined by its degrees of freedom and when the degrees of freedom increase the t distirbution becomes a normal distribution approximately.  

The degrees of freedom represent "the number of independent observations in a set of data. For example if we estimate a mean score from a single sample, the number of independent observations would be equal to the sample size minus one."

Solution to the problem

For n=6

In order to find the critical value we need to take in count that we are conducting a two tailed test, so we are looking on the t distribution with df=n-1=6-1=5 degrees of freedom, a value that accumulates of the area on each tail. We can use excel or a table to find it, for example the code in Excel is:  

"=T.INV(1-0.025,5)", and we got the critical values given by:  

For n=12  

In order to find the critical value we need to take in count that we are conducting a two tailed test, so we are looking on the t distribution with df=n-1=12-1=11 degrees of freedom, a value that accumulates of the area on each tail. We can use excel or a table to find it, for example the code in Excel is:  

"=T.INV(1-0.025,11)", and we got the critical values given by:  

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.