Sampling Distributions and the Central Limit Theorem

Today we worked through some sample problems using the normal model. I also showed students how to find probabilities using their calculators and how to find a z-score from a percentile on their calculator.

Next, we looked at the idea of sampling distributions and the Central Limit Theorem (CLT). This took about 45 minutes to go through and hopefully established some foundation of the concept and form of sampling distribution models. Next class we'll work through some problems and then review for the upcoming mid-term.

Below is the outline of the lesson with italicized comments enclosed in square brackets [like this].

—Sample Distribution of Sample Mean

·         Take random samples of 3 students and calculate mean height [Took 7 random samples and compared means to the class mean.]

   o   What happens? [Did any sample have a mean the same as the class? Were the means all the same?]

   o   Sampling error – each sample yields a different mean [Each sample differs, this is sample variability.]

·         What would you expect to happen if took larger samples? [Students thought spread would decrease; continue to increase sample until sample entire population then how much variation-none!]

   o   Mean stays the same but the spread decreases

·         What will the mean of the means be? [Mean will be close to population mean, more samples of larger size provides even closer results.]

   o   Population mean

·         What will the standard deviation be?

   o   Population standard deviation / sqrt(n) [Students realize variability decreases, just now how.]

·         What is distribution of the sample means?

   o   Create a histogram [Created histogram of the seven sample means, unimodal, slightly skewed but symmetric enough-call this nearly normal. A normal model can be useful in helping to understand behavior.]

·         CLT

   o   Discuss mean and standard deviation of sampling distribution

·         A look at sampling distributions [Showed different sample populations, graphs of large number of samples for various sizes; all look normally distributed but the standard deviation gets smaller.]

·         Calculate for siblings

   o   Use sample size of 5 then 10 then 20 to demonstrate law of diminishing returns [Started by having students calculate population (the class) mean and standard deviation. Showed that the sampling distribution model would be of the form N( mean, std dev / sqrt(sample size) ). Next asked students to calculate the sampling distribution standard deviation for a sample size of 5, then 10, then 20. Showed that doubling size each time has less and less effect on decreasing the size of  the  sampling model standard deviation. This is the law of diminishing returns. Students wondered why the standard deviation of the sampling model was so much smaller than the standard deviation of the population. It was a good opportunity to remind students we are looking at the distribution of means calculated from drawn samples. It turns out that the standard deviation of this sampling model is connected to the standard deviation of the population but as the samples we select increase in size the variation we see in the means of the samples decrease until we sample the entire population. At this point every sample will be the same and there will be no sample variation in the sample means.]

z-scores and the standard normal model

Today focused on finding z-scores and working with the standard normal model. This lesson went pretty much as planned. The only thing I didn't cover was how to find normal probabilities on the calculator. We'll do this at the start of next class. The news that the calculator could calculate values was greeted by good-natured groans and curses.

Below is an outline of the lesson with italicized comments enclosed in square brackets [like this].

—Normal Model

·         Which is more unusual?

   o   Discuss as class, how could these be compared? [Compare someone walking into the room who is 82" tall or with a show size of 4.]

·         Quick investigation [Went as described, allows to see students where z-score is derived.]

   o   Pick 5 different values

   o   Calculate the mean and standard deviation

   o   Now, add a value of 9 to each original value

      §  What happens to mean and standard deviation

   o   Next, subtract a value of 9 from each original value

      §  What happens to mean and standard deviation

   o   Multiply each original value by 9

      §  What happens to mean and standard deviation

   o   Divide each original value by 9

      §  What happens to mean and standard deviation

·         What will the mean and standard deviation be if you subtract x-bar from each datum

   o   Mean is zero, standard deviation unchanged

·         What will the mean be if you divide transformed data by the standard deviation

   o   Mean is zero and standard deviation one

   o   What are the units of this transformed data [Students need to realize that z-scores are unitless.]

   o   This is known as a z-score

·         What is the meaning of the z-score

   o   Tells how many standard deviations a value is away from the mean

·         Going back to opening question, what if we calculate the z-score for both items? [Didn't have data, used their 5 numbers that they picked.]

   o   Can compare to see which is more unusual

·         Work with z-scores on worksheet

·         The distribution of z-scores is known as the standard normal model [Used 15 z-scores from their randomly selected numbers and created a histogram. Students described distribution as unimodal with a slight left skew.]

   o   Mean is zero and standard deviation is one

·         Calculate areas under curve [Worked through several examples using a z-table.]

   o   Using table

   o   Using 68.26, 95.44, 99.74 rule

·         Practice with normal model worksheet

More on Probability Models and Expected Values

Today was a wrap up of probability models and expected value. We were in a computer lab today, so I spent some time on using Minitab as students begin to gather and analyze data for their projects. I was hoping to get into normal models today as well but that will have to wait until next week.

I used roulette as an example to have students create probability models and calculate expected values. I like using this since the probability models are simple and it turns out that all of the expected values are the same, regardless of whether you be on a specific number, bet on even or odd, bet on red or black, bet on the 2nd 12 numbers, bet on the center column; every expected value turns out to be the same. This does two things: 1) it provides easy practice for students to create a probability model and calculate the expected value and 2) makes students understand that a game like roulette is designed based on probabilities and expected values. The expectation from the casino's perspective is that you will lose. This is how casinos can build ostentatious hotels and provide customers free rooms and free drinks.

After, we took a look at a specific probability model, the binomial model. This model is generated from Bernoulli trials, which we defined. A Bernoulli trial has three characteristics:

1. There are fixed probabilities.
2. There are only two possible outcomes for each trial.
3. Each trial is independent.

For a binomial model, we are examining the number of successful outcomes for Bernoulli trials for any given amount of trials. For example, the probability of tossing 7 heads in 10 coin tosses.

After examining a binomial model situation we went into how to calculate the probabilities. With the other non-topic items that were being covered today that was as much as we accomplished.

Below is an outline of today's lesson with italicized comments enclosed in square brackets [like this].

·         Roulette expected values [had a student pick a number and then used this to calculate different expected values based on the color and number picked, i.e. red, even, 1-18, etc.]

·         Coin flipping questions

   o   Have students calculate probabilities

   o   What are characteristics of these problems [posed this question and students came up with all three requirements.]

      §  Two possible outcomes

      §  Constant probability

      §  Independent trials

      §  These are Bernoulli trials

      §  Number of successes in fixed trials is binomial

   o   How did you go about calculating these probabilities?

      §  Success/failures and number of ways

      §  Give binomial formula and explain

      §  Discuss how to calculate with a calculator [Used a baseball player getting 4 at bats and possessing a .300 batting average. What is the probability of getting exactly 2 hits and what is the probability of getting at least 3 hits.]

   o   What is the mean and standard deviation for a binomial model [Asked class how many heads they would expect to toss in 10 coin tosses. Discussed how the obtained 5 and then went into formulas. Showed the standard deviation only to reinforce that a probability model possesses both a mean and a standard deviation.]

      §  np and sqrt(npq)

Bayes Theorem, Probability Models, and Expected Value

Based on where things were left off, I thought a little more work with Bayes Theorem was warranted. But first, we had to finish discussing the Birthday problem. Most of the class hadn't made any headway on the problem, although a couple had calculated the probabilities for up to seven people in the room.

I went through the calculations, which resulted in

    P(no one with the same birthday in a room of 30 people) = 364! / (365²⁹ x 336!)

It's a nice formula but most calculators and spreadsheets get an overflow error trying to calculate values this large. I demonstrated how lists could be made, manipulated and used to calculate the result. It works out to approximately a 70% probability. This is surprisingly large; most people think the value would be much smaller. This lies at the heart of statistical analysis; should we be shocked by something we observe or is it really something that is expected. In this case, with such a high probability, it is not shocking that at least two people in a room of 30 have the same birthday.

From there, we worked with another Bayes problem involving jumping paper frogs. The frogs are indistinguishable but one frog lands on its feet more often than the other. We pick a frog and it lands on its feet. What is the probability that we picked the one that lands on its feet the most often. Most students did a good job creating a tree diagram for the problem, and with minor prompting most were able to determine the desired "reversed" probability.

I then asked what the probability would be if the frog we selected landed on its feet twice in a row. This led to some discussion as to whether or not the probability would change; some thought it would be the same. After calculating the probabilities, they saw that the probability grew larger that the better jumper was selected. This makes sense; every time we see another feet-first landing our confidence we feel more confident that we picked the better jumper.

After, we looked at probability models. I used simple examples like rolling a single die or flipping a coin. I then asked students to consider what the average of roles would be if we rolled a single die numerous times. They were unclear, so I simulated 100 die rolls and calculated a sample mean of 3.58. I then asked how many times we should expect to roll a 1 if we tossed a die 1200 times, and how many 2's etc. After calculating the total point values and dividing by the number of rolls, we came to a result of 3.5. I repeated the question but as what the average would be if we rolled 12,000 times and 12,000,000 times. The expectation is that the average should be 3.5. This is not a value you can roll but what we expect the average of all our rolls to become.

We worked through a problem involving sibling counts and the percentage occurrence of each. I asked students to calculate the expected number of siblings. One student suggested multiplying the value times the percentage. Referring back to the die roll, the class saw that this worked. I then used this to define expected value. We practiced on a simple coin tossing game: two heads wins $1, one head and one tail wins $0.50, and two tails loses $1.

Below is the outline of what was covered today along with italicized comments enclosed in square brackets [like this].

o   Bayes

§  Jumping frogs

o   The Birthday problem

§  Results involve factorials and permutations 

—Discrete Random Variables

·         What is a random variable?

o   Variable that depends on chance [discussed concept briefly]

·         What is a discrete random variable?

o   Can list out all possible values of variable [used die roll as an example]

·         When make a table and histogram of a variable it is the probability distribution and probability histogram

o   Use dice roll example [made histogram on calculator after simulating; with thousands of rolls would expect histogram to flatten out showing a uniform distribution]

·         What is the mean dice roll?

o   This is called the expected value [expected value and mean of the probability model are equivalent terms]

o   Use sibling example to calculate the expected value

o   What does this value mean?

·         What is the standard deviation of a dice roll?

o   Consider how standard deviation was calculated before [showed how to calculate on a calculator and then showed formula]

o   What does this value mean? [as with previous standard deviation, the average distance from the mean]

Conditional Probability and Independence

The ideas of conditional probability and independence were covered today. Using a table, it's relatively straight-forward to demonstrate what conditional probability is. I tried a new demonstration using card guessing and hints to help illustrate independence. Ultimately, I used the definition of P(A | B) = P(A) for independence, then went into the general multiplication rule P(A) x P(B) = P(A | B) x P(B) to show that an equivalent demonstration of independence would be P(A) x P(B) = P(A and B).

These ideas are the easy part. Going into Bayes Theorem is much more confusing. I like to use tree diagrams and work through determining the probability that we are interested in and then focusing on the probability that is of interest out of that total amount. I tell students to not get hung up on the formula but to consider what we are interested in calculating. The tree diagrams help to sort this out and, as probabilities are calculated for the different branches, we ultimately perform the calculations used in Bayes Theorem without getting hung up on what pieces fits where within the formula.

I wrapped up class with the birthday problem. This illustrates how quickly a probability calculation can become messy and for students to think about calculating the compliment of the situation.

Below is an outline of today's lesson along with italicized comments enclosed in square brackets, [like this]

·         Conditional Probability and Independence

o   M and Ms

   §  P(orange| peanut) [referenced back to the original M and M problem to kick things off]

o   Asthma and smoking

   §  Table

   §  Venn Diagram [did not demonstrate this today, will reference it tomorrow]

   §  Tree [used the Three Strings tree diagram to illustrate this]

o   Card probabilities

   §  Ask for volunteer

   §  Choose card from deck; student can win $1 if guesses drawn card (no suit necessary); student writes guess on slip of paper which is not shared with anyone

   §  Ask class “what is the probability of having a correct guess?” answer 4/52

   §  For a penny give a hint—card is red (or black)

   §  “what is the probability of a correct guess?” answer 2/26

   §  For another penny another hint—card is a heart (or appropriate suit)

   §  “what is the probability of a correct guess?” answer 1/13

   §  For another penny give one more hint—card is a number (or face card)

   §  “what is the probability of having a correct guess?” answer 1/9 or ¼

   §  Discuss what information was helpful and what wasn’t

   §  Talk about independence informally [this went as described, really had students think about what information influenced any change in guess]

o   Independence

   §  Go back through problems to check on independence

      ·         M and Ms [asked if being orange and type were independent, had students demonstrate using both definitions]

      ·         Roulette [didn't use]

      ·         Asthma [students looked at whether living with a smoker and having asthma were independent]

      ·         Clothing and gender [didn't use]

o   Bayes

   §  P(plain | orange) [students calculated and then showed tree diagram]

   §  Asthma [didn't use, looked at medical testing results problem from book]

   §  Clothing (shorts or jeans, etc depending on what you see in class) and gender [didn't use, may do this next class]

·         Counting and probability
o   The Birthday problem

   §  Have students record day of birth in month column [turned out we had four pairs of same birthday, one of which had three individuals]

   §  Pose question [the question being is it unusual to have a matched birthday or is it something we should expect]

   §  Results involve factorials and permutations [after students worked on the problem a bit, discussed the direct calculation, building from two students to three to four; then talked about calculating no matching birthdays, where the result is much easier to calculate; this was homework, along with working  through some Bayes Theorem problems]

Investigating Probability

The focus today was on looking at probability. I looked at basic probability properties and rules for calculation. Many of these were covered last class but I did introduce mutually exclusive events. I also discussed the concept of independence informally. I used roulette to illustrate many of the properties and rules for calculation.

Afterward, I tried an investigation I saw at the NCTM annual conference. You have 3 strings and you are blindly tying ends together. The possible outcomes are to form one large loop, one medium and one small loop, or three small loops. The class actively discussed what the possible outcomes were and how to identify or count the outcomes. As expected, students tried to apply formulas they had learned in the past without considering the problem situation. I used a tree diagram (a new way to represent probability situations) and we worked through the probabilities. The results were counter to the class's intuition as to what result should be most likely. I liked this activity; it illustrated many components of the basic probability properties and rules and helped show how tree diagrams are created and used for calculating probabilities.

Below is the outline of the class along with commentary that is enclosed in square brackets and italicized, [like this].

·         Reference basic probability rules

·         Roulette practice

o   P(red)

o   P(even)

o   P(3^rd 12)

o   P(1-18 or center column) [did not reference general addition rule here, but noted how we accounted for numbers that were both 1-18 and in center column]

o   P(2^nd 12 and 3^rd column)

o   P(black and even)

o   P(black or even) [now referenced general addition rule]

o   P(not 19-36)

o   P(0 or 00)

o   P(0 and 00)

o   P(1^st spin red and 2^nd spin black) [spins are independent so that probabilities are multiplied]

o   Is the event  “8 and black” mutually exclusive

o   Is the event “25 and black” mutually exclusive

·         Strings

o   Take 3 strings

o   Grab and fold in half

o   Swirl to randomize

o   Tie 2 ends together, do it again, tie last pair together

o   What are possible outcomes [as discussed above, outcomes are: 1 loop, 2 loops, or 3 loops]

o   Which do you think most likely to occur, which least likely to occur [class guessed that 2 loops would be most common]

o   Tabulate results and represent [broke student work to have students share thinking on how they were configuring outcomes; needed to do this as some students were getting themselves off track with thoughts like there are 6 ends being matched with each other so 6 x 6 = 36 total outcomes]

     §  What are actual probabilities, show using trees [worked through a branch and then had students attempt to complete other branches, most got stuck]

A First Look at Probability

Today we undertook looking at probability. I usually start probability off by asking students to think about different properties they may have learned about probability over the years. I use a think-pair-share format. This helps get out what students already know about the topic and helps to uncover misconceptions that need to be addressed.

Normally, I have students that know something about probability, even if it is rather rudimentary. Today, virtually  no one knew anything about probability, other than possessing some misconceptions. It was good to learn that there was little or no prior knowledge about probability and also to address some misconceptions, such as if you flip a coin several times in a row and they all are heads that the probability of flipping a tail increases or that the more unlikely an event the greater the probability that it occurs in a large number of trials.

I covered some basic ideas and used a bowl full of M&Ms with 50 plain and 100 peanut M&Ms as an example. The class was told that 12% of the plain and 15% of the peanut M&Ms were orange. With this information I asked several different questions about the bowl and the probability of their occurrence. I also used the class characteristics of male and female versus those wearing shorts and those not wearing shorts.

Next class will be focused on specific probability rules, representing probabilities, and practicing their use.

Below is the outline of today's lesson, with comments enclosed in square brackets and italicized [like this].

Probability Concepts

·         What probability rules do you know?

      o   Think, pair, share

      o   Run through basic concepts

        §  0 ≤ P(A) ≤1

        §  P(A) = 0 means never happens

        §  P(A) = 1 means always happens

·         How can you represent probability situations

      o   Tables [used coin toss to illustrate a table]

      o   Venn diagrams [used gender and wearing/not wearing shorts to illustrate]

      o   Use M&M problem

·         Calculating probabilities

      o   P(peanut) [discussed how calculated to bring in idea of (desirable outcomes)/(total outcomes)]

      o   P(not plain)

      o   P(orange and plain)

      o   P(orange or plain)

      o   P(orange)