# We drew some curves to demonstrate how a t distribution
# differs from a normal distribution.

# Here's a grid of values to provide the X-axis of the plot:

> x <- seq(-5,5,.01)
> head(x)
[1] -5.00 -4.99 -4.98 -4.97 -4.96 -4.95

# Here's a standard normal curve. (That's "ell," not
# "one" in the "type='l'" subcommand.)

> par(pin=c(6,4))
> plot(x, dnorm(x), type='l')
> abline(h=0)

# Here, we add a t distribution with small degrees
# of freedom. Notice the heavier tails:

> lines(x, dt(x, df=2), lty=2)

# Because of the heavier tails, the critical value of
# a test statistic with t(2) as the reference distribution
# will be much larger than the 1.96 we saw for a 2-tailed
# Z test at alpha=.05:

> qt(.975, 2)
[1] 4.302653

# As degrees of freedom increase, the curve becomes
# more similar to the standard normal curve:

> lines(x, dt(x, df=14), lty=3)

# And the critical value becomes smaller (but is still
# higher than 1.96):

> qt(.975, 14)
[1] 2.144787

# Here are the estimates of elapsed time we collected
# in class last time:

> seconds
 [1] 20 14 25 18 15 23 30 13 40 12 25 32 30 12
> mean(seconds)
[1] 22.07143
> sd(seconds)
[1] 8.704237

# Because we don't KNOW sigma, we have to estimate the
# standard error of the mean using the sample standard deviation:

> se <- sd(seconds) / sqrt(length(seconds))
> se
[1] 2.326305

# We calculate the one-sample t statistic:

> t <- (mean(seconds) - 23) / se
> t
[1] -0.3991615

# We can evaluate that by comparing it to the critical value;
# it's smaller, so the result is non-significant:

> length(seconds)
[1] 14
> qt(.975, 13)
[1] 2.160369
 
# Here's the probability that a t statistic would be lower than
# ours if the true mean is 23 seconds:

> pt(t, 13)
[1] 0.348127

# But we have to consider also the probability that it's just as
# extreme in the other direction, so here's the probability of
# being this extreme in absolute value:

> 2*pt(-abs(t), 13)
[1] 0.6962541

# 0.696 is larger than our alpha level of .05, so we reach the
# same conclusion that we did using the critical value: non-significant.
# (The conclusion based on comparison of the statistic with its critical
# value will ALWAYS be the same as the one based on comparison of the
# exact p-value to the alpha level.)
 

# We evaluate the normality of the distribution:

> stem(seconds)

  The decimal point is 1 digit(s) to the right of the |

  1 | 223458
  2 | 0355
  3 | 002
  4 | 0

> qqnorm(seconds); qqline(seconds)

# The non-standardized effect size is about -.9 seconds:

> mean(seconds)-23
[1] -0.9285714

# The standardized effect size is about -.1 standard deviations:

> (mean(seconds)-23)/sd(seconds)
[1] -0.1066804
 
# Here's a standardized effect size comparing the Peabody
# mean to a theoretical value of 70:

> attach(JackStatlab)
The following object is masked _by_ .GlobalEnv:

    X

> (mean(CTPEA)-70)/sd(CTPEA)
[1] 0.8026107
 
# Unlike the example with seconds, an unstandardized effect
# size would not be especially useful here because the metric
# of Peabody is not intrinsically meaningful:

> (mean(CTPEA)-70)
[1] 10.02