2. Two sample quantitative analysis

Mastery Quiz Prep

Decoding the problem

Confidence Intervals

  • Confidence interval
    • Interpret: I am _(percent)_% confident that _(bigger group)_ _(has/succeeds)_ an average of _(low end of interval)_ to _(high end of interval)_ _(response variable units)_ more than _(smaller group)_..
    • For example: I am 95% confident that dogs weigh an average of 13.3 to 22.5 pounds more than cats.
    • OR plus-minus form: I am 95% confident that dogs weigh an average of 17.9 pounds more than cats with a margin or error of ±4.6 pounds.

Hypothesis Tests

Answer the following for each of the problems below:
a)    What is the explanatory variable in this scenario. What are its options (treatments)?
b)    What is measured for each individual (the response variable in this scenario)? Is it quantitative or categorical?
c)    If it is quantitative, is there matched pairs or two distinct samples?
d)    Based on the last two responses, what type of interval/test will you perform in StatKey?
e)    What is the null hypothesis in this scenario? Use the correct symbols (μ or p) and use subscripts so it is clear what each symbol means.
f)    What is the alternative hypothesis in this scenario?  Re-read the problem to see if there is an intended direction.
g)    What is the p-value of your test?
h)    Is this an observational study or experiment?  Based on this and your p-value, what can you conclude?
i)    What is the estimated difference between the two groups (95% interval)?
j)    Convert this interval to plus-minus form.
k)    Interpret the confidence interval of the difference in a sentence.  Use plus-minus form because it is often far more readable in a sentence.

1. A group of disc golfers wants to compare two different putting techniques.  In one, the player throws the disc with the forehand, just like a normal Frisbee.  In the other, the player uses a backhand flick to throw.  To test the methods, they get 20 volunteers to throw 8 discs each from 30 feet away.  Half of the volunteers are randomly assigned to one of the techniques, and the rest to the other, before the throwing starts.  The group wants to prove that throwing forehand will work better on average than throwing backhand.  Results (number of shots made out of 8):
    Backhand: 2, 7, 3, 8, 3, 2, 1, 4, 3, 4
    Forehand: 3, 3, 7, 6, 8, 4, 5, 6, 4, 8

2. A group of robot enthusiasts who were not impressed with the frisbee shooters made two t-shirt launchers.  The first cannon, the larger one, has a 3" diameter barrel.  The smaller one has a 2" barrel.  A pile of 20 t-shirts is randomly divided between between the two cannons and their launch distances (in feet) are recorded.  The group is convinced that the large cannon is better.
    Small cannon trials: 80, 56, 78, 61, 31, 64, 72, 66, 69, 78
    Large cannon trials: 84, 97, 88, 77, 91, 83, 43, 89, 79, 67

3. A student group decided to compare how well players did using two different strategies of building a card tower.  The subjects were first instructed on a specific method they needed to use for their tower and told them it was required to use this strategy.  Since the group didn’t want players to mix strategies, they tested two completely separate groups of people.  People volunteered to play and were randomly assigned to a strategy using a coin flip on the day of the experiment.  The results:
    Strategy 1 (seconds required to build the tower): 33, 42, 59, 68, 73, 91, 33, 45
    Strategy 2 (seconds required to build the tower): 73, 33, 49, 62, 65, 48, 47, 66

Free Response Prep

Explain how StatKey simulates tests of independence for matched-pairs data (same question, all versions of quiz):
Discuss explanation in class.  Since the null hypothesis is that treatment group does not affect response, It mixes up the pairing of treatment group and the responses randomly for each bootstrap sample.  For each sample, it takes the average of group A and group B, subtracts, and plots the difference on the big graph distribution.  After thousands of repeats, you get a probability distribution of possible values.  Your p-value is the probability of finding data as high/low/extreme as the difference of your sample averages

Practice solutions

1. Frisbee #1:
        a) Explanatory Variable: Method of throwing (categorical)
            Treatments: Forehand, backhand (the options of the categorical)
        b) Number of shots made (quantitative)
        c) 2 distinct samples
        d) Test for a difference in means.  There is one categorical variable and one quantitative variable.  The reason it is a difference in means is because there are two groups (the options) and each has a quantitative value associated with it.  You take the difference of these means.
        e)   H0: μf = μb
        f)    HA: μf > μb
           g) p= 0.054 (depending on your randomly generated data, it could be above or below .05)
        h) Experiment; We cannot reject the null because our p-value was greater than p= .05. We can get a bigger sample size and 
            collect more data to try and get a better p-value. 
        i) -3.30 to 0.10
        j) -1.60 ± 1.70 more backhand  (1.60 ± 1.70 more forehand)
        j) I am 95% confident that Frisbee players make an average of 1.6 ± 1.7 more shots with their forehand than their backhand. 

2. T-shirt launcher:
        a) Explanatory Variable: Size of the cannon, 
            Treatment Groups: 3" Diameter or 2" Diameter
        b) Distance of the t-shirt; It is quantitative
        c) 2 distinct samples
        d) Test for a difference in means.  There is one categorical variable and one quantitative variable.  The reason it is a difference in means is because there are two groups (the options) and each has a quantitative value associated with it.  You take the difference of these means.
        e)   H0: μs = μL
        f)    HA: μ< μL
           g) p= .021
        h) p-value is statistically significant and we can reject the null
        i) -26.60 to -1.80
        j) -14.2 ± 12.4
        k) I am 95% confident that the 3" cannon shoots 14.2 ± 12.4 feet further than the 2" cannon.  (Note that since I knew the 3" was the further one from the data, I ignored the negative sign on the confidence interval and went directly into writing a logical sentence).

3. Card tower:
  • a) Explanatory Variable: Method Used
    Treatment Groups: Strategy 1 and Strategy 2 
  • b) time required to build the tower; quantitative
  • c) 2 Distinct Samples
  • d) 1 Categorical, 1 Quantitative: There are two groups, a quantitative response variable, and no matched pairs
  • e) Null: the time required to build the tower with the two different strategies is the same
    H0: μ1 = μ2
  • f) Alt: the time required to build the tower with the two different strategies is different (because we don't know which one is supposed to be "better")
        HA: μ1 ≠ μ2
  • g) p = 0.989
  • h) Experiment: 2 different groups (and their only difference is the strategy used), volunteers are randomly assigned to each group
     The data suggests that the 2 strategies are nearly identical.  There is not a shred of evidence to support that they are different in the student population at this school.
  • i) Roughly -15.5 to 16.5
  • j) .5 ± 16
  • k) I am 95% confident that Strategy 1 takes 0.5 ± 16 more seconds to complete than Strategy 2.


compare mu1-mu2 = 0 to mu(1-2) = 0 instead of just mu1=mu2
changing dance partners, but two non-mixing groups, nobody gets left out or replicated