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 | 180 | 125 |
Sample standard deviation | 17.1 | 22.9 | 12.5 |
(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
(μ_{1} − μ_{2}) = 0?
Yes, the interval in part (a) contains (μ_{1} − μ_{2}) = 0.Yes, the interval in part (b) contains (μ_{1} − μ_{2}) = 0. Yes, both intervals contain (μ_{1} − μ_{2}) = 0.No, neither interval contains (μ_{1} − μ_{2}) = 0.
Why is this of interest to the researcher?
If (μ_{1} − μ_{2}) = 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 (μ_{1} − μ_{2}) = 0 is contained in the confidence interval, it is implied that there was an error in the database records. If (μ_{1} − μ_{2}) = 0 is contained in the confidence interval, it is implied that the room rate for one of the hotels was $0.If (μ_{1} − μ_{2}) = 0 is contained in the confidence interval, it is implied that the average room rate for the two hotels was $0.If (μ_{1} − μ_{2}) = 0 is contained in the confidence interval, it is implied that there is a difference in the average room rates for the two hotels.
(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.
a)
z value at 95% = 1.96
CI = ( x1 -x2)+/- z *sqrt(s1^2/n1 + s2^2/n2)
= ( 135 - 125) +/- 1.96 *sqrt(17.1^2/50 + 12.5^2/50)
= (4.13,15.87)
The 95% CI is 4.13 to 15.87
b)
z value at 99% = 2.576
CI = ( x1 -x2)+/- z *sqrt(s1^2/n1 + s2^2/n2)
= ( 180 - 125) +/- 2.576*sqrt(22.9^2/50 + 12.5^2/50)
= (45.50,64.50)
The 95% CI is 45.50 to 64.50
c)
No, neither interval contains (μ1 − μ2) = 0.
d)
Yes, 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 B and the Hotel C chains
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