An article reported that for a sample of 52 kitchens with gas cooking appliances monitored during a one-week period, the sample mean CO2 level (ppm) was 654.16, and the sample standard deviation was 165.4. Required:
a. Calculate and interpret a 9596 (two-sided) confidence interval for true average C02 level in the population of all homes from which the sample was selected.
b. Suppose the investigators had made a rough guess of 175 for the value of s before collecting data. What sample size would be necessary to obtain an interval width of 50 ppm for a confidence level of 95%?

a)    

b)

So the answer for this case would be n=189 rounded up to the nearest integer

Step-by-step explanation:

Part a

represent the sample mean

population mean (variable of interest)

s=165.4 represent the sample standard deviation

n =52represent the sample size  

The confidence interval for the mean is given by the following formula:

  (1)

The degrees of freedom aregiven by:

Since the Confidence is 0.95 or 95%, the significance and , and the critical value would be

Now we have everything in order to replace into formula (1):

Part b

The margin of error is given by this formula:

   (a)

And on this case we have that ME =25 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

  (b)

The critical value for this case wuld be , replacing into formula (b) we got:

So the answer for this case would be n=189 rounded up to the nearest integer

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3