Question

Concerns about climate change and CO2 reduction have initiated the commercial production of blends of biodiesel (e.g., from renewable sources) and petrodiesel (from fossil fuel). Random samples of 32 blended fuels are tested in a lab to ascertain the bio/total carbon ratio.

(a) If the true mean is .9370 with a standard deviation of 0.0090, within what interval will 99 percent of the sample means fall? (Round your answers to 4 decimal places.)

The interval is from____ to ____

(b) If the true mean is .9370 with a standard deviation of 0.0090, what is the sampling distribution of X¯¯¯ ?

1.Exactly normal with μ = .9370 and σ = 0.0090.

2.Approximately normal with μ = .9370 and σ = 0.0090.

3.Exactly normal with μ = .9370 and σx¯= 0.0090/32−−√.

4.Approximately normal with μ = .9370 and σx¯= 0.0090/32−−√.

(c) What theorem did you use to answer part (b)?

Central Limit Theorem

Chebyshev's Theorem

Pythagorean Theorem

Law of Large Numbers

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 0.9370

sample standard deviation = s = 0.0090

sample size = n = 32

Degrees of freedom = df = n - 1 = 32 -1 = 31

a) At 99% confidence level

= 1-0.99% =1-0.99 =0.01

/2 =0.01/ 2= 0.005

t/2,df = t_{0.005,31 = 2.74}

t _/2,df = 2.74

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

= 2.74* (0.0090 / 32)

Margin of error = E = 0.00437

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

- E < <  + E

0.9370 - 0.00437 < < 0.9370 + 0.00437

0.9326 < < 0.9414

(0.9326,0.9414)

The interval is from 0.9326 to 0.9414

b)

n = 32

= 0.9370

= / n = 0.0090/ 32 = 0.00159

4.Approximately normal with μ = .9370 and σx¯= 0.0090/32−−√.

c) Central Limit Theorem