Question

"Durable press" cotton fabrics are treated to improve their recovery from wrinkles after washing "Wrinkle recovery angle" measures how well a fabric recovers from wrinkles. Higher is better. Here are data on the wrinkle recovery angle (in degrees) for the some fabric specimens.
Permafresh Hylite
124 147
104 199
142 149
111 156
123

A consumer group suspects that there is a difference in recovery from wrinkles after washing. Specifically, they want to test the claim that there is a difference in recovery between using Permafresh and Hilite. To investigate this, they identified the mean recovery from wrinkles after washing by measuring the wrinkle recovery angle in degrees for a sample of fabrics using Permafresh, and a sample of fabrics using Hylite.

At the 5% level of significance, is there enough evidence to conclude that there is a difference in wrinkle recovery angle between Permafresh and Hylite?

It is often reasonable to conclude that measurements are normal in distribution. So it is reasonable to assume the population of wrinkle recovery angle is normal for both Permafresh and Hylite. Also assume that the data represents a SRS (simple random sample) of fabrics using Permafresh and Hylite.

a) What are the sample means ? Use 4 decimal places.
Permafresh:
Hylite:

b) What are the sample standard deviations? Use 4 decimal places.
Permafresh:
Hylite:

Answer 1

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= $\frac{1}{(n₁-1)}$[(∑x₁²i)-(∑x₁i)²/n₁] = $\frac{1}{4}$[(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= $\frac{1}{(n₂-1)}$[(∑x₂²i)-(∑x₂i)²/n₂] = $\frac{1}{3}$[(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= (X[bar]₁ - X[bar]₂) - (μ₁ - μ₂) ~t$_{n_1+n_2-2}$

                Sa√(1/n₁+1/n₂)

Sa²= (n₁-1)S₁² + (n₂-1)S₂² = (4*210.7)+(3*598.92) = 377.08

                n₁ + n₂ - 2                     5+4-2

Sa= 19.418≅ 19.42

t= (X[bar]₁ - X[bar]₂) - (μ₁ - μ₂) =    (120.8-162.75) - 0 = -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:

$t_{n_1+n_2-2; \alpha/2 } = t_{7; 0.025 } = -2.365$

$t_{n_1+n_2-2; 1- \alpha/2 } = t_{7; 0.975 } = 2.365$

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!