• The standard error
  • how it differs from the standard deviation
  • when it is used
• The normal distribution
  • computing probabilities and quantiles using pnorm() and qnorm()
• Other distributions (poisson, uniform)
• the mean, the median and the mode
• Confidence intervals

• From calculating probabilities to a statistical test
• Understanding test statistics
• Null and alternative hypotheses
• Understanding and interpreting p-values
• The two types of errors we can commit in a statistical test (type I and type II errors)
• Practice questions for the midsemester test

If we want a more quantitative statement on this question

• We need to know whether our sample is ‘unusual’ or ‘normal’?
• We need to know what ‘unusual’ or ‘normal’ means, so
  • we need to quantify what is usual, normal, rare etc.!
• We need a testable hypothesis

So let’s try

Our hypothesis is:

• BIOL501 students are NOT different from the average New Zealand body height (this is our so-called null hypothesis \(H_0\))
• Because we can stick with this hypothesis or reject it, we need an alternative hypothesis \(H_A\): BIOL501 students ARE different from a typical sample of New Zealanders
• Note that the Null hypothesis is negative, which makes it easier to falsify!
• Also note that we can never accept \(H_0\), we can only fail to reject \(H_0\)

Now we need a test statistic and knowledge of the distribution we compare against:

Our test statistic is simply the mean of our sample (let’s do it for females to start with):

mean(d1$bodyheight[d1$sex == 'F']) 
[1] 163.3333

How does that compare with

pop = rnorm(2500000, mean = 164, sd = 6) #why 2500000?
mean(pop)
[1] 164.0014

YES!

We can tell by just looking at how we compare to the distribution of NZ female bodyheight. We could conclude that we are typical New Zealanders. More quantitatively…:

The probability of obtaining a value equal or smaller than the mean we got for our female students when sampling from the NZ population is about 50%:

pnorm(q = mean(d1$bodyheight[d1$sex == 'F']) , mean = 164, sd = 6)
[1] 0.4557641

In other words our sample is NOT unusual and hence we cannot reject our null hypothesis!

We are NOT different from a typical sample of New Zealanders

OK, but what if our sample mean had been different, say 160 cm, or 150 cm…?

pnorm(q = 160, mean = 164, sd = 6)
[1] 0.2524925

Is a value that we’d get 25% of the time by chance rare?

pnorm(q = 150, mean = 164, sd = 6)
[1] 0.009815329

Is a value that we’d get 1% of the time by chance rare?

• Those probabilities (50%, 25%, 1%) are our p-values. They always are to be interpreted as ‘the probability of obtaining such an extreme value by chance’
• We need a threshold to objectively distinguish from ‘rare’ and ‘unusually rare’:
  • Normally, we say that if our sample is more extreme than what we would find 5% of the time by chance, then our p-value is significant (in a statistical sense)
  • This threshold is also called the \(\alpha\)-threshold
• In a report, we would for instance say (if the mean was 150 cm): ‘The body height of our female students is significantly lower than the New Zealand average (p < 0.05)’ (but see restrictions on next slide…!)

In all of this, we can make 2 types of errors!

• A type I error is when we falsly reject the null hypothesis \(H_0\)
  • In plain language, this means that we call something ‘significant’ (e.g. a difference, 2 samples, etc.) while in reality there is no significant difference (or, more generally ‘nothing going on’)
• A type II error is when we falsly fail to reject the null hypothesis \(H_0\)
  • In plain language, this means that “we don’t see anything where in reality things (e.g. samples) are different”

Note that the ‘plain language’ definitions are inexact, but hopefully help you to understand the principle of type I/II errors

Maybe easier to remember…:

The variance is

1. A measure for the spread in a sample
2. The sum of squared errors divided by the degrees of freedom
3. The squared standard deviation
4. Can be calculated using the function var() in R
5. All of the above

The standard error of the mean

1. Is just another way of expressing the spread in a sample
2. Is a way of showing how good our estimate of the mean is
3. Is the squareroot of the standard deviation
4. All of the above
5. None of the above

Without using R, what is the probability of being taller than 170 if you are a male and we assume that the body height of the male population follows a perfect normal distribution with mean 170?

1. Close to 100%
2. About 25%
3. 50%
4. 45%
5. It is impossible to calculate this because you don’t indicate the standard deviation

Without turning to R, what is the approximate body height that 75% of women are under (shorter than), assuming a population mean of about 160 cm and a standard deviation of about 6?

1. About 165 cm
2. About 195 cm
3. Certainly less than 160 cm
4. About 155 cm
5. None of the above

A histogram of the body heights of the female students in this paper

1. Will look rather bell-shaped
2. Will look like a normal distribution with a mean around 160
3. Will have a mode that is approximately equal to the mean and the median
4. Will likely have a standard deviation between 5 and 10
5. All of the above is correct

Without using R or a calculator, what is the approximate probability to draw a random number larger than 8 from a normal distribution with mean 4 and standard deviation 1?

1. About 50%
2. About 10%
3. Less than 5%
4. Very small, less than 1%

If you were to show the effect of a binomial variable on a continuous variable, which of the following plots would suit best?

1. A histogram
2. A box plot
3. Such a data set cannot be visualised
4. None of the above

What is the reason for dividing by \(n-1\) rather than \(n\) when calculating the variance of a sample?

1. To avoid division by zero
2. To make sure the sum of squared residuals (deviances) is not zero
3. To avoid underestimating the variance in small sample sizes
4. To make sure the variance has the same units as the original variable

To numerically describe Auckland house prices, which set of metrics would you use?

1. The mean and the range
2. Only the minimum and the maximum
3. The median and perhaps the first and third quartile
4. The mode and the maximum
5. None of the above

Which of the following is not adding non-systematic variation?

1. If the experimenter becomes tired and starts making random mistakes
2. If a cheaper, less accurate machine is bought to analyse the samples
3. If patients are by mistake administered too low a dosis of a medication
4. If suddenly the experimental subjects are taken from a more heterogeneous population

If you were commissioned to estimate the mean size of snapper in the Hauraki Gulf (your population), a good sample to take would be

1. 200 snapper from Waiheke island
2. 200 snapper taken from random spots around the Hauraki Gulf
3. 200 snapper from random places all around New Zealand
4. 500 snapper randomly taken from around the Hauraki Gulf

If the sample size increases

1. So does the standard error
2. The standard error remains the same
3. The standard error decreases
4. The variance increases
5. The standard deviation increases

Which line of code provides the answer to the question ‘What is the probability of being between 185 and 200 cm tall for a male of a population with mean body height 175 cm and standard deviation of 7?’

1. pnorm(q = 200, mean = 175, sd = 7) - pnorm(q = 185, mean = 175, sd = 7)
2. pnorm(q = 185, mean = 175, sd = 7) - pnorm(q = 200, mean = 175, sd = 7)
3. pnorm(q = 200, mean = 185, sd = 7) - pnorm(q = 185, mean = 175, sd = 7)
4. pnorm(q = 200, mean = 175, sd = 7) + pnorm(q = 185, mean = 175, sd = 7)
5. None of the above

Imagine you are researching the size of great white sharks around New Zealand waters. Assume this variable is normally distributed with mean 3 m and standard deviation 0.5 m.

1. Sketch such a distribution, label all axes and indicate the mean and the standard deviation.
2. At least how big are the biggest 5% of your population of sharks? Shade this probability in your sketch.
3. You realise that in actual fact, great white shark size does not follow a normal distribution. The actual distribution is heavily skewed to the right. Sketch such a distribution and show how your estimate from (2) might have been terribly wrong.

• To formulate a null and an alternative hypothesis
• How to conduct a very simple statistical test
• What a test statistic is, an example for this
• What a type I and a type II error is

Sounds like not much, but if you understand this, you’re set!

• test statistic
• null hypothesis \(H_0\)
• alternative hypothesis \(H_A\)
• \(\alpha\)-level
• significance
• type I error
• type II error