Answer:  y + 1 = 2 (x -1)
Step-by-step explanation:
Point slope form is:
y - y1 = m (x - x1)
Point on the line is (1, -1) and slope is 2
m = 2
y1 = -1
x1 = 1
y - (-1) = 2 (x - 1) or y + 1 = 2 (x -1)
 
        
             
        
        
        
Answer:
a. We reject the null hypothesis at the significance level of 0.05
b. The p-value is zero for practical applications
c. (-0.0225, -0.0375)
Step-by-step explanation:
Let the bottles from machine 1 be the first population and the bottles from machine 2 be the second population.  
Then we have  ,
,  ,
,  and
 and  ,
,  ,
,  . The pooled estimate is given by
. The pooled estimate is given by  
 
a. We want to test  vs
 vs  (two-tailed alternative).
 (two-tailed alternative).  
The test statistic is  and the observed value is
 and the observed value is  . T has a Student's t distribution with 20 + 25 - 2 = 43 df.
. T has a Student's t distribution with 20 + 25 - 2 = 43 df.
The rejection region is given by RR = {t | t < -2.0167 or t > 2.0167} where -2.0167 and 2.0167 are the 2.5th and 97.5th quantiles of the Student's t distribution with 43 df respectively. Because the observed value  falls inside RR, we reject the null hypothesis at the significance level of 0.05
 falls inside RR, we reject the null hypothesis at the significance level of 0.05
b. The p-value for this test is given by  0 (4.359564e-10) because we have a two-tailed alternative. Here T has a t distribution with 43 df.
0 (4.359564e-10) because we have a two-tailed alternative. Here T has a t distribution with 43 df.
c. The 95% confidence interval for the true mean difference is given by (if the samples are independent)
 , i.e.,
, i.e.,
 
where  is the 2.5th quantile of the t distribution with (25+20-2) = 43 degrees of freedom. So
 is the 2.5th quantile of the t distribution with (25+20-2) = 43 degrees of freedom. So
 , i.e.,
, i.e.,
(-0.0225, -0.0375)
 
        
             
        
        
        
Answer:
b. circle; 
Step-by-step explanation:
The given conic has equation;

We complete the square to obtain;

This is a circle with center;

This implies that;

When the circle is rotated through an angle of  ,
,
The new center is obtained using;
 and
 and 
We plug in the given angle with x and y values to get;
 and
 and 
This gives us;

The equation of the rotated circle is;

Expand;

Multiply through by 4; to get

Write in general form;

Divide through by 2.

 
        
             
        
        
        
If you want to check if your answer is correct, expanding it all out will help!
(x-6)(x+2)=0 <-- Use FOIL tactic 
x²+2x-6x-12=0
x²-4x-12=0 
As you can see, this is identical to the initial equation given. The x values would be 6 and -2, as seen from the factored equation. 
Hope I helped :)