# Data:  30 randomly selected daily high temperatures from 
# summer days in New York City (May through September)

summer.temps <- c(59,81,85,75,81,77,92,83,92,76,74,73,57,68,80,69,
83,78,87,88,93,71,73,76,81,82,73,64,80,79)

mean(summer.temps)
sd(summer.temps)


# Test whether mean temperature is greater than 75:
t.test(summer.temps, alternative="greater", mu=75)


# Test whether mean temperature is different from 75:
t.test(summer.temps, alternative="two.sided", mu=75)


# 90% t confidence interval for population mean temperature:
t.test(summer.temps, alternative="two.sided", conf.level=0.90)

# Just the 90% confidence limits:
t.test(summer.temps, conf.level=0.90)$conf.int

#######
#
# Calculating test statistic and P-value
# when you just have the summary statistics:
#

y.bar <- 77.66667
s <- 8.872015
samp.size <- 30

# the hypothesized value of mu:
mu.0 <- 75  

# Calculating t* for a given y-bar, s, and n,:
t.star <- (y.bar - mu.0)/(s/sqrt(samp.size))

# P-value for each possible type of alternative hypothesis:
two.sided.P.value <- 2*(pt(abs(t.star), df = samp.size-1, lower.tail=FALSE))
print(paste("P-value for two-sided test is ", two.sided.P.value))

less.than.P.value <- pt(t.star, df = samp.size-1, lower.tail=TRUE)
print(paste("P-value for less-than alternative is ", less.than.P.value))

greater.than.P.value <- pt(t.star, df = samp.size-1, lower.tail=FALSE)
print(paste("P-value for greater-than alternative is ", greater.than.P.value))

### 90% CI for mu:
conf.level <- 0.90;
alpha <- 1 - conf.level
LCL <- y.bar - qt(1-alpha/2, df = samp.size-1)*(s/sqrt(samp.size)) 
UCL <- y.bar + qt(1-alpha/2, df = samp.size-1)*(s/sqrt(samp.size)) 
print(paste(conf.level*100, " percent CI for mu is ", LCL, UCL))