Question

1. "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 same fabric specimens discussed in the previous exercise:

Permafresh	138	135	132	134	132
Hylite	141	144	147	139	145

The mean (±0.01) and the standard deviation (±0.0001) for the Permafresh are:   
x x¯ = s = .

The mean (±0.01) and the standard deviation (±0.0001) for the Hylite are:   
x¯¯¯x¯ = s = .

The standard error and the value of the t statistic are (±0.001):   
    SE =     t =

The P-value for this two sided test is (±0.01): ⩽P⩽

2.

Do women and men differ in how they perceive their life expectancy? A researcher asked a sample of men and women to indicate their life expectancy. This was compared with values from actuarial tables, and the relative percent difference was computed (perceived life expectancy minus life expectancy from actuarial tables was divided by life expectancy from actuarial tables and converted to a percent). Here are the relative percent differences for all men and women over the age of 70 in the sample:

Men	-26	-24	-19	-20	-14	-12
Women	-23	-17	-15	-12	-11	-7	-8

Use technology to approximate the t distribution for this test. Do NOT use the conservative approach.

The t test statistic (±±0.001) is

The degrees of freedom (±±0.001) of the t test statistic is

Answer 1

Here we will use the technology. R software, the r-codes provided below

Q1)

> Permafresh=c(138,135,132,134,132)
> Hylite=c(141,144,147,139,145)
> Permafreshmean=mean(Permafresh)
> Permafreshmean
[1] 134.2
> sdpermafresh=sqrt(var(Permafresh))
> sdpermafresh
[1] 2.48998
> Hylitemean=mean(Hylite)
> Hylitemean
[1] 143.2
> sdhylite=sqrt(var(Hylite))
> sdhylite
[1] 3.193744
> t.test(Permafresh,Hylite,paired=T)

   Paired t-test

data: Permafresh and Hylite
t = -3.9468, df = 4, p-value = 0.01686
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-15.331269 -2.668731
sample estimates:
mean of the differences
-9

Conclusion: Here we will use the paired t- test because here are data on the wrinkle recovery angle (in degrees) for the same fabric specimens. The sample mean and sample standard deviation for Permafresh is 13.4 and 2.4899. P- value of two sided test is 0.01686 which is less than 0.05% level of significnace, we can conclude that recovery angle of Permafresh and hylite is different.

Q2)

> men=c(-26,-24,-19,-20,-14,-12)
> women=c(-23,-17,-15,-12,-11,-7,-8)
> t.test(men,women,var.equal=T)

   Two Sample t-test

data: men and women
t = -1.9176, df = 11, p-value = 0.08149
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-12.6310685 0.8691637
sample estimates:
mean of x mean of y
-19.16667 -13.28571

Conclusion: The t test statistic is -1.9176 with degrees of freedom 11 . The pvalue (0.08149) is greater than 0.05 (5%) level of significance, we conclude that the women and men may not differ in how they perceive their life expectancy.