. Air entrainment is a process of entrapping tiny air bubbles in concrete mix in order to increase the durability of the hardened concrete in freeze-thaw climates. After 35 days, the breaking stress [in psi] was measured for concrete samples with and without air entrainment. Based on the data, does the air entrainment process increase the breaking stress of the concrete

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Air entrainment is a process of entrapping tiny air bubbles in concrete mix in order to increase the durability of the hardened concrete in freeze-thaw climates. After 35 days, the breaking stress [in psi] was measured for concrete samples with and without air entrainment. Based on the data, does the air entrainment process increase the breaking stress of the concrete. { ∝ = 0.05 }, assuming population variances are equal.

Air Entrainment       No Air Entrainment

4479                                  4118

4436                                  4531

4358                                  4315

4724                                   4237

4414                                   3888

4358                                  4279

4487                                   4311

3984

4197

4327

Since p-value ( 0.088) > significance level ( 0.05)

hence, Failed to reject Null hypothesis

It is then concluded that the null hypothesis H₀ is NOT REJECTED.

Therefore, there is no sufficient evidence to claim that population mean μ1 is greater than μ2 at 0.05 significance level.

We conclude that Air entrainment process can't increase the breaking stress of the concrete.

Explanation:

Given the data in the question;  

mean x" = (4479 + 4436 + 4358 + 4724 + 4414 + 4358 + 4487 + 3984 + 4197 + 4327) / 10

mean x"1 =  43764 / 10 = 4376.4

 x                 ( x - x" )             ( x - x" )²

4479              102.6              10526.76

4436              59.6               3552.16

4358             -18.4                 338.56

4724              347.6               120825.76

4414               37.6                 1413.76

4358             -18.4                 338.56

4487              110.6                 12232.36

3984            -382.4                153977.76

4197              -179.4                32184.36

4327             -49.4                  2440.36  

∑                                             337830.4

Standard deviation s1 = √( (∑( x - x" )²) / n -1  

Standard deviation s1 = √( 337830.4 / (10 - 1 ))

Standard deviation s1 = 193.74

 x2                 ( x2 - x"2 )           ( x2 - x"2 )²

4118                  -121.9               14859.61  

4531                   291.1               84739.21

4315                   75.1                5640.01

4237                  -2.9                 8.41    

3888                  -351.9            123833.61

4279                   39.1               1528.81

4311                     71.1                5055.21

∑                                              235664.87

mean x"2 = (4118 + 4531 + 4315 + 4237 + 3888 + 4279 + 4311) / 7

mean x"2 = 29679 / 7 = 4239.9  

Standard deviation s2 = √( (∑( x2 - x" )²) / n2 - 1  

Standard deviation s1 = √( 337830.4 / (7 - 1 ))

Standard deviation s1 = 198.19

so

Mean x"1 = 4376.4,   S.D1 = 193.74,  n1 = 10

Mean x"2 = 4239.9,   S.D2 = 198.19,   n2 = 7

so;

Null Hypothesis H₀ : μ1 = μ2

Alternative Hypothesis H₁ : μ1 > μ2

Lets determine our rejection region;

based on the data provided. the significance level ∝ = 0.005

with degree of freedom DF = n1 + n2 - 2 = 10 + 7 - 2 = 15

so, Critical Value = 1.753

The rejection region for this right -tailed is R = t:t > 1.753

Test statistics

since it is assumed that the population variances are equal, so we calculate pooled standard deviation;

Sp = √{ [ (n1 -1)S.D1² +  (n2 - 1)S.D2²] / [ n1 + n2 -2 ]

we substitute

Sp = √{ [ (10 -1)(193.74)² +  (7 - 1)(198.19)²] / [ 10 + 7 -2 ]

Sp = √ [ 573492.345 / 15 ]

Sp = 195.53

so the Test statistics will be;

t = (x"1 - x"2) / Sp√( + )

t = (4376.4 - 4239.9) / 195.53√( + )

t = 136.5 / 96.36

t = 1.42

so

P-value = 0.088

Since p-value ( 0.088) > significance level ( 0.05)

hence, Failed to reject Null hypothesis

It is then concluded that the null hypothesis H₀ is NOT REJECTED.

Therefore, there is no sufficient evidence to claim that population mean μ1 is greater than μ2 at 0.05 significance level.

We conclude that Air entrainment process can't increase the breaking stress of the concrete.