1. The amount of time that it takes to complete a certain job is known to be Normally distributed with a mean of 10 minutes and a standard deviation of 1 minute.

curve(dnorm(x,10,1),0,20)

pnorm(11,10,1)
## [1] 0.8413447

There is an 84% chance of finishing in less than 11 minutes.

2. A critical measurement on the diameter of a part that is used in a subassembly is assumed to have a mean of 10mm. The management would like to test this hypothesis against the alternative that it is not equal to 10mm at an alpha= 0.05 level of significance. Towards this end, they have collected a random sample of n=100 parts and measured their diameter.

dataq2<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/diameter.csv",header=TRUE,na.strings="")
Histogram
hist(dataq2$Diameter,main="Parts Measurements",xlab="Diameter",ylab="Count",col="orange")

Boxplot
boxplot(dataq2$Diameter, main="Parts Measurements")

  • State the null and alternative hypothesis, perform the test, and state conclusions.

\(H_0\): Mu = 10
\(H_a\): Mu ≠ 10

t.test(dataq2$Diameter, mu=10, alternative ="two.sided", conf.level = 0.05)
## 
##  One Sample t-test
## 
## data:  dataq2$Diameter
## t = 7.6839, df = 99, p-value = 1.134e-11
## alternative hypothesis: true mean is not equal to 10
## 5 percent confidence interval:
##  10.16899 10.17177
## sample estimates:
## mean of x 
##  10.17038

At the 5% significance level, we can reject the null hypothesis that the true mean is 10 (p-value < 0.05)

3. Researchers at a textile production facility would like to test the hypothesis that the mean breaking strength of abraided fabric is different than that of unabraided fabric at an alpha=0.10 level of significance. Towards this end, they conducted an experiment in which they measured the breaking force of 8 samples of each type of fabric. Assume the populations are approximately Normally distributed and use a two-sample t-test with a pooled variance.

dataq3<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/Fabric.csv",header=TRUE,na.strings="")
boxplot(dataq3$ï..Abraided,dataq3$Unabraided, main="Collected measurements", names=c("Abraided","Unabraided"),ylab="")

t.test(dataq3$ï..Abraided,dataq3$Unabraided,var.equal = TRUE, conf.level= 0.10)
## 
##  Two Sample t-test
## 
## data:  dataq3$ï..Abraided and dataq3$Unabraided
## t = -1.3729, df = 14, p-value = 0.1914
## alternative hypothesis: true difference in means is not equal to 0
## 10 percent confidence interval:
##  -7.871082 -6.528918
## sample estimates:
## mean of x mean of y 
##    36.375    43.575

At the 10% significance level, we dont reject the null hypothesis that the abraided fabric is different than that of unabraided fabric. (p-value < 0.10)

4. Consider a designed experiment in which the crop yield was measured at 2 levels of crop density/spacing (1=dense, 2=sparse) and 3 levels of fertilizer (1=typeA, 2=typeB, 3=typeC). A total of 96 observations were collected. A colleague of yours did some preliminary analysis of the data in R using the following code

dat<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/cropdata2.csv")
str(dat)
## 'data.frame':    96 obs. of  3 variables:
##  $ density   : int  1 2 1 2 1 2 1 2 1 2 ...
##  $ fertilizer: int  1 1 1 1 1 1 1 1 1 1 ...
##  $ yield     : num  177 178 176 178 177 ...
dat$density<-as.fixed(dat$density)
dat$fertilizer<-as.fixed(dat$fertilizer)
interaction.plot(dat$fertilizer,dat$density,dat$yield)

mod<-lm(yield~density+fertilizer+density*fertilizer,dat)
gad(mod)
## Analysis of Variance Table
## 
## Response: yield
##                    Df  Sum Sq Mean Sq F value    Pr(>F)    
## density             1  5.1217  5.1217 15.1945 0.0001864 ***
## fertilizer          2  6.0680  3.0340  9.0011 0.0002732 ***
## density:fertilizer  2  0.4278  0.2139  0.6346 0.5325001    
## Residual           90 30.3367  0.3371                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod<-lm(yield~density+fertilizer,dat)
gad(mod)
## Analysis of Variance Table
## 
## Response: yield
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## density     1  5.1217  5.1217 15.3162 0.0001741 ***
## fertilizer  2  6.0680  3.0340  9.0731 0.0002533 ***
## Residual   92 30.7645  0.3344                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

5. Consider designing an experiment in which we wish to test whether there is a difference in the mean between 4 levels of a single factor (i.e. between 4 populations). Specifically, this is to be a Completely Randomized Design that will be analyzed using ANOVA. We would like to collect a sufficient number of samples such that the test with an alpha=0.05 level of significance would be able to detect with a power of 85% a mean difference that is 50% of the standard deviation.

pwr.anova.test(k=4,n=NULL,f=0.5,sig.level=0.05, power=0.85)
## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 4
##               n = 13.32146
##               f = 0.5
##       sig.level = 0.05
##           power = 0.85
## 
## NOTE: n is number in each group

The sample for each level will be 14 (n=14)

trt1 <- c("lvl1","lvl2","lvl3","lvl4")
design<-design.crd(trt=trt1,r=14, seed=6814224)
design$book
##    plots  r trt1
## 1    101  1 lvl2
## 2    102  1 lvl1
## 3    103  2 lvl1
## 4    104  2 lvl2
## 5    105  3 lvl2
## 6    106  4 lvl2
## 7    107  3 lvl1
## 8    108  1 lvl3
## 9    109  2 lvl3
## 10   110  3 lvl3
## 11   111  1 lvl4
## 12   112  4 lvl3
## 13   113  5 lvl3
## 14   114  5 lvl2
## 15   115  6 lvl3
## 16   116  6 lvl2
## 17   117  7 lvl2
## 18   118  4 lvl1
## 19   119  5 lvl1
## 20   120  2 lvl4
## 21   121  3 lvl4
## 22   122  6 lvl1
## 23   123  7 lvl1
## 24   124  4 lvl4
## 25   125  8 lvl1
## 26   126  7 lvl3
## 27   127  8 lvl3
## 28   128  9 lvl1
## 29   129 10 lvl1
## 30   130  5 lvl4
## 31   131  8 lvl2
## 32   132  9 lvl3
## 33   133  9 lvl2
## 34   134  6 lvl4
## 35   135 10 lvl3
## 36   136 10 lvl2
## 37   137  7 lvl4
## 38   138  8 lvl4
## 39   139  9 lvl4
## 40   140 11 lvl3
## 41   141 10 lvl4
## 42   142 11 lvl1
## 43   143 11 lvl4
## 44   144 12 lvl1
## 45   145 11 lvl2
## 46   146 13 lvl1
## 47   147 12 lvl3
## 48   148 12 lvl2
## 49   149 13 lvl2
## 50   150 12 lvl4
## 51   151 13 lvl4
## 52   152 14 lvl1
## 53   153 13 lvl3
## 54   154 14 lvl2
## 55   155 14 lvl4
## 56   156 14 lvl3