Question

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.

Answer 1

Solution :

Given that,

a ) The sample mean = = 980.2 grams

s = 27.6 grams.

n = 11

a ) Degrees of freedom = df = n - 1 = 11 - 1 = 10

At 99% confidence level the t is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

_{t /2  df} = t_0.005,10 = 3.169

b ) The standard error = (s /n)   

= ( 27.6 /  11 )

= 8.322

c ) Margin of error = E = t_/2,df * (s /n)   

= 3.169 * (27.6 / 11)

= 26.372

Margin of error = E =26.372

D ) The 99% confidence interval estimate of the population mean is,

- E < < + E

980.2 - 26.370 < < 980.2 + 26.370

953.83 < < 1006.57

(953.83,1006.57 )