The coaching staff of professional football team believes th

The coaching staff of professional football team believes that the rushing offense has become increasingly potent in recent years. To investigate this belief, 20 randomly selected games from one year’s schedule were compared to 11 randomly selected games from the schedule five years later. The sample information on passing yards per game (pypg) is summarized below.

Construct the 95% confidence interval for the difference in the population means based on these data. Test, at the 5% level of significance, whether the data on passing yards per game provide sufficient evidence to conclude that the passing offense has become more potent in recent years.

Solution

a)
CI = x1 - x2 ± t a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
Where,
x1 = Mean of Sample 1, x2 = Mean of sample2
sd1 = SD of Sample 1, sd2 = SD of sample2
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
Mean(x1)=203
Standard deviation( sd1 )=38
Sample Size(n1)=20
Mean(x2)=232
Standard deviation( sd2 )=33
Sample Size(n1)=11
CI = [ ( 203-232) ±t a/2 * Sqrt( 1444/20+1089/11)]
= [ (-29) ± t a/2 * Sqrt( 171.2) ]
= [ (-29) ± 2.228 * Sqrt( 171.2) ]
= [-58.15 , 0.15]

b)
t Test of Difference Means for Unequal Variance
Set Up Hypothesis
Null Hypothesis , There Is No-Significance between them Ho: u1 = u2
Alternate, passing offense has become more potent in recent years - H1: u1 < u2
Test Statistic
X(Mean)=203
Standard Deviation(s.d1)=38 ; Number(n1)=20
Y(Mean)=232
Standard Deviation(s.d2)=33; Number(n2)=11
we use Test Statistic (t) = (X-Y)/Sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =203-232/Sqrt((1444/20)+(1089/11))
to =-2.22
| to | =2.22
Critical Value
The Value of |t | with Min (n1-1, n2-1) i.e 10 d.f is 1.812
We got |to| = 2.21639 & | t | = 1.812
Make Decision
Hence Value of | to | > | t | and Here we Reject Ho
P-Value:Left Tail - Ha : ( P < -2.2164 ) = 0.0255
Hence Value of P0.05 > 0.0255,Here we Reject Ho
sufficient evidence to conclude that the passing offense has become more potent in recent years.