Help with statistics? I need to calculate the Confidence Interval (CI) Hotel Rat
ID: 3176392 • Letter: H
Question
Help with statistics?
I need to calculate the Confidence Interval (CI)
Hotel Rating Scenario: Suppose you are a hotel manager whose goal is maintaining the hotel rating at 4-star on average. Recently, you have heard bad rumors about your hotel, so your concern is that the average rating is below 4. Your co-worker collected a sample of n = 100 subjects who rated the hotel. The original data format should be like below. However, your co-worker summarized the data as below and trashed the original data. Star1 2 3 4 5 Total Count 0 6 25 42 27 100 Objective: Let mu denote the average rating in the population. Your goal is to estimate mu. Calculate a 95% CI for mu. Calculate a 99% CI for mu. Interpret the resulting 99% CI.Explanation / Answer
Here the sample size is large (>30) so we can use z-distribution to find the confidence interval. The confidence interval formula is:
X bar (-/+)E
X bar = sample mean
E = margin of error
E = zc * (s / Ön)
zc = critical value
s = sample standard deviation
n = sample size
From normal table we get zc for 95% , 99% are respectively 1.96 and 2.58.
x
f
xf
1
0
0
2
6
12
3
25
75
4
42
168
5
27
135
Total
100
390
x bar = S xf / Sf = 390 / 100 = 3.9
s = Ö[ f * (x-x ba)^2 / (sum of f – 1)]
x
f
xf
x - x bar
(x - x bar)^2
f*(x - x bar)^2
1
0
0
-2.9
8.41
0
2
6
12
-1.9
3.61
21.66
3
25
75
-0.9
0.81
20.25
4
42
168
0.1
0.01
0.42
5
27
135
1.1
1.21
32.67
Total =
100
Total =
75
s = Ö(75/99) = 0.8704
( a )
X bar (-/+)E
= 3.9 (-/+) [ 1.96*(0.8704/Ö100]
= 3.9 (-/+) 0.1706
= 3.729 and 4.071
The 95% confidence interval is (3.729 and 4.071)
Or
The 95% confidence interval is (3.73 and 4.07)
( b )
X bar (-/+)E
= 3.9 (-/+) [ 2.58*(0.8704/Ö100]
= 3.9 (-/+) 0.2246
=3.675 and 4.125
The 99% confidence interval is (3.675 and 4.125)
Or
The 99% confidence interval is (3.68 and 4.12).
( c )
The 99% confidence interval says that here one can be 99% confident that the population mean rating will line in between (3.68 and 4.12)
x
f
xf
1
0
0
2
6
12
3
25
75
4
42
168
5
27
135
Total
100
390
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