A study has been done to test the effect of rock music on auditory attention. Eighteen eighth-grade students are randomly divided into two groups of 9 students, and the same hearing test was given to these students. The first group of students listened to rock music for 25 minutes right before taking the test, and the other group of students took the test without listening to the music. Here are their scores. Group with music: 78 85 87 93 97 80 94 88 86
Group without music: 64 66 77 84 63 72 85 79 80
Assume normal populations with equal variance and construct a 95% confidence interval for the mean difference.
Find the lower bound of the confidence interval (round off to the nearest integer).

The Lower bound of the confidence interval to the nearest integer  =  6

Step-by-step explanation:

Calculate  the Lower bound of the confidence interval

First step : Calculate the mean and standard deviation for each groups

Group 1 ( group with music )

N = 9

Total ( ∑x ) = 788

mean = 788 / 9 = 87.55

Variance ( s^2 ) = 39.777

s = = 6.31

Group 2 ( group without music )

N = 9

∑x = 670

mean = 670 / 9 = 74.44

variance ( s^2 ) = 72.27

s =  = 8.50

Next : Calculate the pooled variance , standard error and confidence interval

Pooled variance ( Sp^2 ) : 896.43 / 16 = 56.03

standard error :  S ( M1 - M2 ) = √((56.03 / 9) + (56.03 / 9)) = 3.53

confidence interval : μ1 - μ2 = ( M1 - M2 ) ± Ts ( M1 - M2 )

                                              = 13.1111  ±  (2.12 * 3.53 )

                                              = ( 5.630 ,  20.591  )

The Lower bound of the confidence interval to the nearest integer  =  6

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