Question

Please type answer, can't see written answers well.

Also PLEASE PROVIDE BELL SHAPED CURVE** That is the part I struggle with the most.

The airlines industry measures fuel efficiency by calculating how many miles one seat can travel, whether occupied or not, on one gallon of jet fuel. The following data show the fuel economy, in miles per seat for 17 randomly selected flights on Delta and United. Assume the two population variances for fuel efficiency for the two airline are equal.

Delta

65.80
81.40
58.90
73.60
53.20
49.80
68.30
61.40
73.10
67.60
72.20
61.00
52.70
71.40
44.90
55.90
86.70

United

82.10
58.80
60.00
57.90
45.20
54.30
68.40
52.00
59.60
63.10
67.40
73.30
77.20
58.00
81.10
88.50
63.00

a. Conduct a the 95% confidence interval estimate of the population?

b. Perform a hypothesis test using α = .05 to determine if the average fuel efficiency  differs between the two airlines.

c. Determine the p-value and interpret the results.

Answer 1

a. Conduct a the 95% confidence interval estimate of the population?

Solution:

Confidence interval for difference between two population means is given as below:

Confidence interval = (X1bar – X2bar) ± t*sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

From given data, we have

Confidence level = 95%

X1bar = 64.58235

X2bar = 65.28824

S1 = 11.3526

S2 = 11.75972

n1 = 17

n2 = 17

df = n1 + n2 – 2 = 17 + 17 – 2 = 32

Critical t value = 2.0369

(by using t-table)

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(17 – 1)* 11.3526^2 + (17 – 1)* 11.75972^2]/(17 + 17 – 2)

S_p² = 133.5863

(X1bar – X2bar) = 64.58235 - 65.28824 = -0.7059

Confidence interval = (X1bar – X2bar) ± t*sqrt[S_p²*((1/n1)+(1/n2))]

Confidence interval = -0.7059 ± 2.0369*sqrt[133.5863*((1/17)+(1/17))]

Confidence interval = -0.7059 ± 2.0369* 3.9643

Confidence interval = -0.7059 ± 8.0751

Lower limit = -0.7059 - 8.0751 = -8.7810

Upper limit = -0.7059 + 8.0751 7.3692

Confidence interval = (-8.7810, 7.3692)

b. Perform a hypothesis test using α = .05 to determine if the average fuel efficiency  differs between the two airlines.

Here, we have to use two sample t test for the difference between two population means assuming equal population variances.

H₀: µ₁ = µ₂ versus H_a: µ₁≠ µ₂

Test statistic formula for pooled variance t test is given as below:

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

We have

S_p² = 133.5863

(X1bar – X2bar) = 64.58235 - 65.28824 = -0.7059

sqrt[S_p²*((1/n1)+(1/n2))] = 3.9643

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

t = -0.7059 / 3.9643

t = -0.1781

We are given

α = 0.05

df = 32

Critical values = -2.0369 and 2.0369

Test statistic = t = -0.1781 is lies within Critical values = -2.0369 and 2.0369

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that the average fuel efficiency differs between the two airlines.

c. Determine the p-value and interpret the results.

We have

Test statistic = t = -0.1781

df = 32

So, P-value by using t-table is given as below:

P-value = 0.8598

α = 0.05

P-value > α = 0.05

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that the average fuel efficiency differs between the two airlines.