n <- 25
df <- (n-1)
me <- ((77-65)/2)
tval <- qt(.95, df)
tval
## [1] 1.710882
MarginError <- (df-me)/2
MarginError
## [1] 9
sample_dev <- ((77+65)/2)
sample_dev
## [1] 71
sd<- (me/tval)*5
sd
## [1] 17.53481

z <- 1.65 
margin <- 25
stand_dev <- 250

Raina <- round(((z*stand_dev )/margin)^2,0)
Raina
## [1] 272

z <- 2.58
me <- 25
stand_dev  <- 250

Luke <- round(((z*stand_dev )/me)^2,0)
Luke
## [1] 666
samp <- 200

There does not seem to be a clear difference in the average reading and writing scores

We would assume that the scores are independant

H0: There is no difference in the average scores in reading and writing.

HA: There is a difference in average scores.

The observations are independent and the distribution is normal with no skew.

diff <- -.545
df <- samp - 1
stand_dev <- 8.887
SE <- stand_dev/sqrt(samp)
t <- (diff-0)/SE
pval <- pt(t, df)
pval
## [1] 0.1934182

Since we did not reject the null hypothesis, we may run into a type II error.

With our conclusion that there is no difference we would expect 0 to be in our confidence interval.

auto_mean <- 16.12
auto_sd <- 3.58
auto_n <- 26
man_mean <- 19.85
man_sd <- 4.51
man_n <- 26
df <- 26-1
meandiff <- auto_mean - man_mean
SEauto <- auto_sd/sqrt(auto_n)
SEman <- man_sd/sqrt(man_n)
SE <- sqrt((SEauto^2)+(SEman^2))
t <- (meandiff-0)/SE
pval <- pt(t, df)
pval <- 2*pval
pval
## [1] 0.002883615

H0: All hours are the same across the five groups

HA: There is one value that is different across all five groups

The survey is random. The sample size is less than 10% of the population. We assume that the events are independant across all groups. The distribution looks normal and variability appears to be about equal.

n <- 1172
g <- 5
meansqd <- 501.54
sumsqR <- 267382
p <- .06812

df1 <- g-1
df2 <- n-g
df3 <- df1 + df2
c(df1, df2, df3)
## [1]    4 1167 1171
sumsq1 <- df1 * meansqd
sumsq2 <- sumsqR + sumsq1
c(sumsq1, sumsq2)
## [1]   2006.16 269388.16
meanR <- sumsqR/df2
meanR
## [1] 229.1191
fval <- meansqd/meanR
fval
## [1] 2.188992

THe p-value is greater than .05 therefore we do not reject the null hypothesis and conclude that there is no significant difference between the 5 groups.