STAT1008 Lecture Notes - Lecture 4: Null Hypothesis, Statistic, Alternative Hypothesis
4 HYPOTHEISS TESTS
4.1 INTRODUCING HYPOTHESIS TESTS
Statistical Test
- A statistical test uses data from a sample to assess a claim about a population
Extrasensory Perception
- Sample proportion = 0.32
Statisitcal Hypotheses
- A statistical test are framed formally in terms of two competing hypotheses:
- Null Hypothesis (Ho): Claim that there is no effect or difference.
- Alternative Hypothesis (Ha): Claim for which we seek evidence.
- The alternative hypothesis is established by
observing evidence (data) that contradicts
the null hypothesis and supports the
alternative hypothesis
- Hypotheses are always about population
parameters.
- ESP Hypotheses
o For the ESP experiment:
o Ho: p = 1/5 <- o effet or o differee
o Ha: p? 1/5 <- Clai e seek eidee for
- Helpful hints
o Ho usually includes =
o Ha usually includes >, <, or ≠
o The inequality in Ha depends on the question
Sleep versus Caffeine
- Students were given words to memorize, then randomly assigned to take either a 90 min nap, or a
caffeine pill. 2 ó hours later, they were tested on their recall ability.
- Explanatory variable: Treatment (sleep or caffeine)
- Response variable: Number of words recalled
- Is there a difference in average word recall between sleep and caffeine?
- Let μs ad μ e the ea uer of words recalled after sleeping and after caffeine.
o H: μs = μ
find more resources at oneclass.com
find more resources at oneclass.com
o HA: μs ≠ μ
- The following hypotheses are equivalent, and either set can be used:
o H: μs – μ =
o HA: μs – μ ≠
Hypotheses
- Write down the hypothesis for each of the following situations;
o Dose the proportion of people who support gun control differe between males and females?
▪ pf : proportion of females who support gun control H0: p f = p m
▪ p m: proportion of males who support gun control Ha : p f ≠ p
o Is the average hours of seep per night for college students less than 7?
▪ μ: aerage hours of sleep per ight for ollege studets
▪ H: μ = Ha: μ <
Your Own Hypothesis
- Come up with a situation where you want to establish a claim based on our class survey data
o What parameter(s) are you interested in?
o What would the null and alternative hypotheses be?
o What type of data would lead you to believe the null hypothesis is probably not true?
Statistical Significance
- When results as extreme as the observed sample statistic are unlikely to occur by random chance alone
(assuming the null hypothesis is true), we say the sample results are statistically significant
- If our sample is statistically significant, we have convincing evidence against H0, in favor of HA
- If our sample is not statistically significant, our test is inconclusive
- Statistical significance is a difficult concept, but also one of the most fundamental concepts of the
course
- We return to this concept almost every class for the rest of semester.
Extrasensory Perception
- p = Proportion of correct guesses
o H0: p = 1/5
o Ha: p > 1/5
- If results are statistiall sigifiat…
o the sample proportion of correct guesses is higher than is likely just by random chance (if ESP does
not exist and p = 1/5)
o we have evidence that the true proportion of correct guesses really is higher than 1/5, and thus
have evidence of ESP
- If results are NOT statistiall sigifiat…
o the sample proportion of correct guesses could easily happen just by random chance (if ESP does
not exist and p = 1/5)
o we do not have enough evidence to conclude that p > 1/5, or that ESP exists
Key Question
- How unusual is it to see a sample statistic as extreme as that observed, if Ho is true?
- If it is very unusual, we have statistically significiant evidence again the null hypothesis
- Today question: How do we measure how unusual a sample statistic is, if Ho is true?
Summary
- Statistical tests use data from a sample to assess a claim about a population
- Statistical tests are usually formalized with competing hypotheses:
- Null hypothesis (H0): no effect or no difference
- Alternative hypothesis (Ha): what we seek evidence for
find more resources at oneclass.com
find more resources at oneclass.com
- If data are statistically significant, we have convincing evidence against the null hypothesis, and in favor
of the alternative
4.2 MEASURING EVIDENCE WITH P-VALUES
Key Question
- How unusual is it to see a sample statistic as extreme as that observed, if Ho is true?
- If it is very unusual, we have statistically significiant evidence again the null hypothesis
- Today question: How do we measure how unusual a sample statistic is, if Ho is true?
Measruing Evidence againg H0
- To see if a statistic provides evidence against H0, we need to see what kind of sample statistics we
would observe, just by random chance,
o if H0 were true
Paul the Octopus
- We need to know what kinds of statistics we would observe just by random chance, if the null
hypothesis were true
- How could we figure this out???
- Simulate many samples of size n = 8 with p = 0.5
Simulate!
- We can simulate this with a coin!
- Each coin flip = a guess between two teams
o (Heads = correct, Tails = incorrect)
- Flip a coin 8 times, count the number of heads, and calculate the sample proportion of heads
- Respond to the poll-everywhere with the number of heads you have got.
- Ho etree is Paul’s saple proportio of ?
- We just created our first randomization distribution!
Randomization Distribution
- A randomization distribution is a collection of statistics from samples simulated assuming the null
hypothesis is true
- The randomization distribution shows what types of statistics would be observed, just by random
chance, if the null hypothesis were true
Lots of simulations
- To estimate the p-value more accurately, we need many
more simulations!
Key Question
- How unusual is it to see a sample statistic as extreme as
that observed, if H0 is true?
- A randomization distribution tells us what kinds of
statistics we would see just by random chance, if the null
hypothesis is true
- This makes it straightforward to assess how extreme the observed statistic is!
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
A statistical test uses data from a sample to assess a claim about a population. A statistical test are framed formally in terms of two competing hypotheses: Null hypothesis (ho): claim that there is no effect or difference. Alternative hypothesis (ha): claim for which we seek evidence. The alternative hypothesis is established by observing evidence (data) that contradicts the null hypothesis and supports the alternative hypothesis. Helpful hints: ho usually includes , ha usually includes >,
