Distillation is a process for separating and collecting substances according to their reaction to heat. when heat is applied to a mixture, the substance that evaporates and is collected as it cools is the distillate. the unevaporated portion of the mixture is the residue. oil obtained from orange blossoms through distillation is used in perfume. suppose the oil yield is normally distributed. in a random sample of eleven distillations, the sample mean oil yield was 980.2 grams with sample standard deviation 27.6 grams. if possible, answer (a) to (d). if not possible, explain.(a) find the point estimate.(b) find the standard error.(c) find the margin of error at the 99% confidence level.(d) find and interpret the 99% confidence interval.

a:  980.2

b:  26.4

c:  2.7%

d:  953.8 < µ < 1006.4

Step-by-step explanation:

We need to construct a 99% confidence interval from our sample data.  The sample size is 11, so we will use t-distribution.  Our degrees of freedom are 10, our t-value we will need is: 3.169

x = 980.2, s = 27.6

We will use these values to construct the confidence interval

See attached photo for the construction of the confidence interval...

A point estimate is the the sample mean of 980.2

The standard error is 26.4 (calculated on the attached photo)

The margin of error is how many percent the confidence interval may be off by.  In this case, it's 26.4/980.2 = 0.0269, or 2.7%

The meaning of the confidence interval tells us that we are 95% sure that the true population mean is within the range of 953.8 to 1006.4 grams

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.