Question

Manufacturers of aircraft engines have, like many other industrial manufacturers, tried to make engines that are more economical to operate while still providing the power needed to fly large-capacity aircraft. The Boeing 747, long considered the workhorse of the jumbo jet fleet, is powered by four engines, each capable of 44,700 lbs of thrust. Because 747’s have been around since 1969, maintenance costs now are relatively fixed; the goal for any airline is to plan for consistent, predictable maintenance costs.

The new era of more efficient planes include the Boeing 777, introduced in 1995, which has been used to replace many 747’s in airline company’s fleets. The 777’s engines feature aviation’s first-ever carbon-fiber composite fan blades, and each engine is capable of 84,000 lbs of thrust.

Not only are the newer engines capable of greater power, their efficiency also should provide lower maintenance cost variance. The following engine maintenance cost data (in $ millions) from a sample of the United Airlines fleet is provided on these two types of aircraft:

Plane type:	747	777
n	101	32
̅x	$15.8	$7.5
s2	$7.29	$2.25
s	$2.70	$1.50

Hint: keep your data truncated as presented; do not convert the data into $millions.

A. Construct a 95% confidence interval for the standard deviation of the 747's engine maintenance cost.

Provide your answers to 4 decimal places.

   to  

B. Construct a 95% confidence interval for the standard deviation of the 777's engine maintenance cost.

Provide your answers to 4 decimal places.

     to  

C. Hypothesis testing. United Airlines now wants to test if the maintenance cost variance of its newer 777 planes is less than the industry standard.

According to data collected by the National Air Transportation Association, the industry standard maintenance cost variance for the fleet of all domestic-based 777 planes is $2.30 (in $ millions).

What is the proper statement of the hypothesis test criteria?

HO: σ2 ≥ ≤ = HA: σ2 < > ≠

D. Using the data from the sample, answer the fill-in-the-blank questions, and make the correct hypothesis test conclusion, at a level of significance α = .10.

Reject Ho if the test statistic of

is

> <

the critical value of

Based on these results, we should:

• Reject Ho

• Accept Ho

E. Given your decision to Accept or Reject the null hypothesis, what does the result of the hypothesis test tell us about the efficiency of United Airlines maintenance costs for it's 777 fleet compared to the industry average cost for all domestic airlines who also operate the 777?

Answer 1

a)

s = 2.7, n = 101

95% Confidence interval for population standard deviation :

Critical value, χ²α/2 = CHISQ.INV.RT(0.05/2, 100) = 129.5612

Critical value, χ²1-α/2 = CHISQ.INV.RT(1-0.05/2, 100) = 74.2219

Lower Bound = √((n-1)s²/χ²α/2) = √((101 - 1)2.7²/129.5612) = 2.3721

Upper Bound = √((n-1)s²/χ²1-α/2) = √((101 - 1)2.7²/74.2219) = 3.1340

2.3721 < σ < 3.1340

b)

s = 1.5, n = 32

95% Confidence interval for population standard deviation :

Critical value, χ²α/2 = CHISQ.INV.RT(0.05/2, 31) = 48.2319

Critical value, χ²1-α/2 = CHISQ.INV.RT(1-0.05/2, 31) = 17.5387

Lower Bound = √((n-1)s²/χ²α/2) = √((32 - 1)1.5²/48.2319) = 1.2026

Upper Bound = √((n-1)s²/χ²1-α/2) = √((32 - 1)1.5²/17.5387) = 1.9942

1.2026 < σ < 1.9942

c)

s² = 2.25, n = 32

Null and alternative hypothesis:

Hₒ : σ² = 2.3

H₁ : σ² < 2.3

d)

Test statistic:

χ² = (n-1)s² / σ² = (32 - 1)2.25/2.3 = 30.3261

Degree of freedom:

Df = n -1 = 32 - 1 = 31

Critical value, χ²α = CHISQ.INV(0.05, 31) = 19.2806

Reject Ho if the test statistic of χ² = 30.3261 is > the critical value of 19.2806.

Decision:

Accept Ho

e)

There is not enough evidence to conclude that the maintenance cost variance of it newer 777 is less than the industry standard.