Question

In: Statistics and Probability

The values listed below are waiting times​ (in minutes) of customers at two different banks. At...

The values listed below are waiting times​ (in minutes) of customers at two different banks. At Bank​ A, customers enter a single waiting line that feeds three teller windows. At Bank​ B, customers may enter any one of three different lines that have formed at three teller windows. Answer the following questions.

Bank A

6.46.4

6.66.6

6.76.7

6.86.8

7.17.1

7.37.3

7.47.4

7.97.9

7.97.9

7.97.9

Bank B

4.14.1

5.35.3

5.85.8

6.26.2

6.66.6

7.87.8

7.87.8

8.48.4

9.39.3

10.010.0

Construct a 99​% confidence interval for the population standard deviation

sigma at Bank A.

_min < σ Bank A < _min

Construct a 99​% confidence interval for the population standard deviation

sigma at Bank B.

_min < σ Bank B < _min

Solutions

Expert Solution

Bank A ( X ) Σ ( Xi- X̅ )2 Bank B ( Y ) Σ ( Yi- Y̅ )2
6.4 0.64 4.1 9.1809
6.6 0.36 5.3 3.3489
6.7 0.25 5.8 1.7689
6.8 0.16 6.2 0.8649
7.1 0.01 6.6 0.2809
7.3 0.01 7.8 0.4489
7.4 0.04 7.8 0.4489
7.9 0.49 8.4 1.6129
7.9 0.49 9.3 4.7089
7.9 0.49 10 8.2369
Total 72 2.94 71.3 30.901

Mean X̅ = Σ Xi / n
X̅ = 72 / 10 = 7.2
Sample Standard deviation SX = √ ( (Xi - X̅ )2 / n - 1 )
SX = √ ( 2.94 / 10 -1 ) = 0.5715

Mean Y̅ = ΣYi / n
Y̅ = 71.3 / 10 = 7.13
Sample Standard deviation SY = √ ( (Yi - Y̅ )2 / n - 1 )
SY = √ ( 30.901 / 10 -1) = 1.853

For Bank A

S2 = 0.3266
α = 0.01
n = 10
((n-1)S2 / χ2 (0.01/2)) < σ2 < ((n-1)S2 / χ2 (1 - 0.01/2) )
(( 10-1 ) 0.3266 / χ2 (0.01/2) ) < σ2 < ((10-1)0.3266 / χ2 (1 - 0.01/2) )
χ2 (0.01/2) = 23.5894
χ2 (1 - 0.01/2) ) = 1.7349
Lower Limit = (( 10-1 ) 0.32662 / χ2 (0.01/2) ) = 0.1246
Upper Limit = (( 10-1 ) 0.32662 / χ2 (0.01/2) ) = 1.6943
99% Confidence interval is ( 0.1246 , 1.6943 )
( 0.1246 < σ2 < 1.6943 )
( 0.353 < σ < 1.3016 )

For Bank B

S2 = 3.4336
α = 0.01
n = 10
((n-1)S2 / χ2 (0.01/2)) < σ2 < ((n-1)S2 / χ2 (1 - 0.01/2) )
(( 10-1 ) 3.4336 / χ2 (0.01/2) ) < σ2 < ((10-1)3.4336 / χ2 (1 - 0.01/2) )
χ2 (0.01/2) = 23.5894
χ2 (1 - 0.01/2) ) = 1.7349
Lower Limit = (( 10-1 ) 3.43362 / χ2 (0.01/2) ) = 1.31
Upper Limit = (( 10-1 ) 3.43362 / χ2 (0.01/2) ) = 17.8122
99% Confidence interval is ( 1.31 , 17.8122 )
( 1.31 < σ2 < 17.8122 )
( 1.1446 < σ < 4.2205 )

orchestra answered 1 year ago
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