#null hypothesis umean >=10000
xbar=9900
umean=10000
sd=120
n=30
z=(xbar-umean)/(sd/sqrt(n))
z
## [1] -4.564355
alpha=0.05
z.alpha=qnorm(1-alpha)
-z.alpha
## [1] -1.644854
#NULL hypothesis is rejected
pnorm(z)
## [1] 2.505166e-06
#null hypothesis <=2 grams is umean
xbar=2.1
umean=2
sd=0.25
n=35
z=(xbar-umean)/(sd/sqrt(n))
z
## [1] 2.366432
pnorm(z,lower.tail = F)
## [1] 0.008980239
alpha=0.05
#Ho is rejected
#null hypothesis weight =15.4 kg is umean
xbar=15.4
umean=14.6
sd=2.5
n=35
z=(xbar-umean)/(sd/sqrt(n))
z
## [1] 1.893146
pval=2*pnorm(z,lower.tail = F)
pval
## [1] 0.05833852
alpha=0.05
#pval>alpha, we accept the null hypothesis
#that no change has occured in penguin weight
#Ho is accepted
#null hypothesis umean >=10000 population sd unknown
xbar=9900
umean=10000
sd=125
n=30
t=(xbar-umean)/(sd/sqrt(n))
t
## [1] -4.38178
alpha=0.05
t.alpha=qt(1-alpha,df=n-1)
-t.alpha
## [1] -1.699127
#NULL hypothesis is rejected
pval=pt(t,df=n-1)
pval
## [1] 7.035026e-05
# >60% voting
pbar=85/148
p0=0.6
n=148
sp=(sqrt(p0*(1-p0)/n))
sp
## [1] 0.04026936
z=(pbar-p0)/sp
z
## [1] -0.6375983
alpha=0.05
z.alpha=qnorm(1-alpha)
-z.alpha
## [1] -1.644854
pval=pnorm(z)
pval
## [1] 0.2618676
#NULL hypothesis is accepted
#power of test and TV watching 6 hours
xbar=
umean=4
sd=2
n=4
(sd/sqrt(n))
## [1] 1
z=(xbar-umean)/(sd/sqrt(n))
z=xbar-4
alpha=0.05
z.alpha=qnorm(1-alpha)
-z.alpha
## [1] -1.644854
xbar=4+z.alpha
xbar
## [1] 5.644854
umean2=6
z2=(xbar-umean2)/(sd/sqrt(n))
#1-beta
beta=pnorm(z2)
beta
## [1] 0.36124
1-pnorm(z2)
## [1] 0.63876
#NULL hypothesis is rejected
pnorm(z)
## [1] 0.5