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Answer:
At 5% there is significant evidence to reject the null hypothesis. You can conclude that there is a difference between the population mean of the wrinkle recovery angle of fabric treated with Permafresh and the population mean of the wrinkle recovery angle of fabric treated with Hylite.
a)
X[bar]₁= 120.8 degrees
X[bar]₂= 162.75 degrees
b)
S₁²= 14.5155 degrees
S₂²= 24.4727 degrees
Step-by-step explanation:
Hello!
The study variable is X: wrinkle recovery angle for a fabric specimen.
There is a suspicion that there is a difference in recovery from wrinkles after washing between two products (Permafresh and Hilite). To test this suspicion two random samples of fabric, 5 were treated with Permafresh and 4 were treated with Hylite resulting in the data:
Sample 1 (Permafresh)
X₁: wrinkle recovery angle for a fabric specimen treated with Permafresh.
X₁~N(μ₁;σ₁²)
n₁= 5
124; 104; 142; 111; 123
Sample mean X[bar]₁= (∑x₁i)/n₁= 604/5= 120.8 degrees
Sample variance S₁²= [(∑x₁²i)-(∑x₁i)²/n₁] =
[(73806)-(604)²/5]= 210.7 degrees²
S₁= 14.515 ≅ 14.52 degrees
Sample 2 (Hylite)
X₂: wrinkle recovery angre for a fabric specimen trated with Hylite.
X₂~N(μ₂;σ₂²)
n₂= 4
147; 199; 149; 156
Sample mean X[bar]₂= (∑x₂i)/n₂= 651/4= 162.75 degrees
Sample variance S₂²= [(∑x₂²i)-(∑x₂i)²/n₂] =
[(107747)-(651)²/4]= 598.916 ≅ 598.92 degrees²
S₂= 24.472≅ 24.47 degrees
To test the suspicion that there is a difference between the winkle recovery angle on fabric samples treated with Permafresh and Hylite, the hypothesis is:
H₀: μ₁ = μ₂
H₁: μ₁ ≠ μ₂
α: 0.05
To test the difference between the population means, considering that only sample information is available and the size of both samples, the most appropriate statistic to use is a pooled t for independent samples (unknown but equal population variances):
t=<u> (X[bar]₁ - X[bar]₂) - (μ₁ - μ₂) </u>~t
Sa√(1/n₁+1/n₂)
Sa²= <u> (n₁-1)S₁² + (n₂-1)S₂² </u> = <u> (4*210.7)+(3*598.92) </u>= 377.08
n₁ + n₂ - 2 5+4-2
Sa= 19.418≅ 19.42
t=<u> (X[bar]₁ - X[bar]₂) - (μ₁ - μ₂) </u>= <u> (120.8-162.75) - 0 </u>= -3.22
Sa√(1/n₁+1/n₂) 19.42√(1/5+1/4)
Using the critical region approach, this rejection region is two-tailed with critical values:
If t ≤ -2.365 or t ≥2.365, then the decision is to reject the null hypothesis.
If -2.365 < t < 2.365, then the decision is to not reject the null hypothesis.
Since the calculated t-value is less than the left critical value, the decision is to reject the null hypothesis.
I've calculated the p-value for the test: 0.0146
This p-value is less than the significance level of 0.05, then, using this approach, the decision is also to reject the null hypothesis.
This means, that at 5% there is significant evidence to reject the null hypothesis. You can conclude that there is a difference between the population mean of the wrinkle recovery angle of fabric treated with Permafresh and the population mean of the wrinkle recovery angle of fabric treated with Hylite.
I hope it helps!