Question

Even within a particular chain of hotels, lodging during the summer months can vary substantially depending on the type of room and the amenities offered. Suppose that we randomly select 50 billing statements from each of the computer databases of the Hotel A, the Hotel B, and the Hotel C chains, and record the nightly room rates. The means and standard deviations for 50 billing statements from each of the computer databases of each of the three hotel chains are given in the table.

	Hotel A	Hotel B	Hotel C
Sample average ($)	135	160	105
Sample standard deviation	17.2	22.2	12.1

(a) Find a 95% confidence interval for the difference in the average room rates for the Hotel A and the Hotel C chains. (Round your answers to two decimal places.)
$  to $  

(b) Find a 99% confidence interval for the difference in the average room rates for the Hotel B and the Hotel C chains. (Round your answers to two decimal places.)
$  to $  

(c) Do the intervals in parts (a) and (b) contain the value (μ₁ − μ₂) = 0?

Yes, the interval in part (a) contains (μ₁ − μ₂) = 0.Yes, the interval in part (b) contains (μ₁ − μ₂) = 0.    Yes, both intervals contain (μ₁ − μ₂) = 0.No, neither interval contains (μ₁ − μ₂) = 0.

Why is this of interest to the researcher?

If (μ₁ − μ₂) = 0 is contained in the confidence interval, it is implied that the room rate for one of the hotels was $0.If (μ₁ − μ₂) = 0 is contained in the confidence interval, it is implied that there is no difference in the average room rates for the two hotels.    If (μ₁ − μ₂) = 0 is contained in the confidence interval, it is implied that there was an error in the database records.If (μ₁ − μ₂) = 0 is contained in the confidence interval, it is implied that there is a difference in the average room rates for the two hotels.If (μ₁ − μ₂) = 0 is contained in the confidence interval, it is implied that the average room rate for the two hotels was $0.

(d) Do the data indicate a difference in the average room rates between the Hotel A and the Hotel C chains?

Yes, the data indicate a difference in the average room rates between the Hotel A and the Hotel C chains.No, the data do not indicate a difference in the average room rates between the Hotel A and the Hotel C chains.    

Do the data indicate a difference in the average room rates between the Hotel B and the Hotel C chains?

Yes, the data indicate a difference in the average room rates between the Hotel B and the Hotel C chains.No, the data do not indicate a difference in the average room rates between the Hotel B and the Hotel C chains.   

Answer 1

a.) We need to find Confidence Interval for difference in mean of Hotel A and C.

Ma = 135

Mc = 105

Standard deviation,

Sa = 17.2

Sc = 12.1

n = 50

Ma - Mc = 135 - 105 = 30

Now, Pooled estimate of variance is given as:

In our case, n1 = n2 = n = 50

So, Sp² = (Sa² + Sc² ) / 2

Sp² = (295.84 + 146.41) / 2

Sp² = 221.125

So, Standard Error of Estimate,

So, SE_(Ma-Mc) =

SE_(Ma-Mc) = 2.974

Degree of freedom, df = 50 + 50 - 2 = 98

So, critical value of t, t = 1.98

Now Confidence Interval is given as:

So, lower limit = (Ma - Mc) - t * SE_(Ma-Mc)

Lower Limit = 30 - 1.98 * 2.974

Lower Limit = 24.11

Upper limit = (Ma - Mc) + t * SE_(Ma-Mc)

Upper Limit = 30 + 1.98 * 2.974

Upper Limit = 35.89

So, CI = ( 24.11, 35.89)

b.)  Here we have to find 99% Confidence Interval for Hotel B and Hotel C

Mb = 160

Mc = 105

Standard deviation,

Sb = 22.2

Sc = 12.1

n = 50

Mb - Mc = 160 - 105 = 55

Now, Pooled estimate of variance is given as:

In our case, n1 = n2 = n = 50

So, Sp² = (Sb² + Sc² ) / 2

Sp² = (492.84 + 146.41) / 2

Sp² = 319.625

So, Standard Error of Estimate,

So, SE_(Mb-Mc) =

SE_(Mb-Mc) = 3.575

Degree of freedom, df = 50 + 50 - 2 = 98

So, critical value of t for 99% Confidence, t = 2.62

Now Confidence Interval is given as:

So, lower limit = (Mb - Mc) - t * SE_(Mb-Mc)

Lower Limit = 55 - 2.62 * 3.575

Lower Limit = 45.634

Upper limit = (Mb - Mc) + t * SE_(Mb-Mc)

Upper Limit = 55 + 2.62 * 3.575

Upper Limit = 64.366

So, CI = ( 45.63, 64.37)

c.)

Correct Option is D. No, neither interval contains (μ1 − μ2) = 0.

If (μ1 − μ2) = 0 is contained in the confidence interval, it is implied that there is no difference in the average room rates for the two hotels. Option B

d.)

Yes, the data indicate a difference in the average room rates between the Hotel A and the Hotel C chains

e.)

Yes, the data indicate a difference in the average room rates between the Hotel B and the Hotel C chains