Problem Set 12 Hints

Firstly, know that EVERY problem MUST have a population parameter (p or mu), sample parameter (x-bar or p-hat), standard deviation (given for means but must be calculated for proportions), and a number of observations which is always n. In some problems it may not be obvious, for example, the problem might use language like "a majority of..." in which case you must think for yourself what a "Majority" looks like as a percentage or as a decimal. Think about it....

Problem One:
The little thing that looks like the Jesus fish is the Greek letter lowercase alpha, which is often used in stat to represent the level of significance. So for example, if alpha = .05 then the problem is using a 5% level of significance.

Problem Two:
Think carefully about which number is which. Remember that your population parameter always comes from the bigger group. NO SHORTCUTS - tell me specifically in CONTEXT what all the assumptions are, and remember that your interpretation is going to require two sentences with the significance test.

Problem Three:
a. I'm looking for a fantastic CUSS here. Label your graph properly. A rough sketch of the histogram is fine but give me an idea of scale. As Toni suggested, starting with "54" probably isn't a good idea - start with something more even.
b. Since the n is so small here, I don't want to hear anything about how we can use the "central limit theorem" in this problem because....we can't. Show WHY we can't and tell me what else we can do (then do that).

Problem Four:
a. use "fewer than half" as a reference for what values you should use for p, your population parameter
b. please do the whole significance testing procedure again for this part
c. DON'T SKIP THIS QUESTION - I know that for about half of you you're going to think about skipping it. It's worth 10 points and will knock you down a letter grade if you skip. Instead, think carefully about what is happening between a and b and tell me what you observe. Write a congratulations to Erin and/or Drew for making state choir after this question for five bonus points. If you get stuck with this question, just ask someone for help.

Problem Five
Do two different significant tests for this question, and think carefully about what "a majority" means. If you got the last question, you'll know what I'm talking about. If not, ask a friend or me for help.

Friday, January 31st: Significance Testing with Means

A significance test (also known as a hypothesis test) compares a sample with a population. Given what we know about the population, is it likely that the sample is representative of the population? Do we have evidence to show that the population parameter may not be accurate? These questions are just a few that statisticians address when performing significance tests.

There are four steps to performing a significance test:
1) State hypotheses (null and alternative) and the population parameter of interest.
Ho: mu = (population number)
Ha: mu >, <, or not = to (population number)
where mu represents.....(whatever your problem is about)

2) State assumptions. They are exactly the same as your confidence interval assumptions (random assumption, size assumption, and independent assumption).

3) Calculate your z-score using the z-score formula from earlier this year:
z = x-bar - mu/Sx/root(n), and calculate the p-value by using the Normalcdf function in your calculator.

4) Interpret in context for both your results and the context of the problem.
-Since the p-value (insert p-value) is (less than/greater than) the significance level (insert sig level here as a decimal), we reject/fail to reject the Null Hypothesis.
*Reject Null if less than, fail to reject Null if greater than
-We do/do not have sufficient evidence to show that (whatever your alternative hypothesis is saying in words, not symbols)

Wednesday, January 29th: Test Review, Thursday January 30th: Confidence Interval Test

We reviewed with two confidence interval problems in class on Wednesday to help reinforce main ideas. I gave you an additional problem to take home with you. Remember that the AP exam people are tricky and that they are going to try to ask you some sort of conceptual problem about the interval along with the calculation of the interval itself.

The tests that I've graded fall somewhere in the middle in terms of test difficulty. It wasn't the hardest test, wasn't the easiest - probably somewhere near the middle. My advice to you guys is this, as I've been saying for a while: try to tackle the open response part FIRST so that you don't run out of time at the end. You can always pick an answer for multiple choice, but it's really tough to make up work at the last minute for the free response.

I will pass back your tests to you as soon as I have everyone take the test. At this moment, I still have a student who needs to make up the exam.

Tuesday January 28th: Confidence Intervals with Small Samples

What happens if your sample size is under 30, and you cannot use the Central Limit Theorem to verify the shape assumption? In this instance, we cannot use the Normal Curve, so we must use a flatter distribution with a wider spread called the t-distribution. The data is more spread out because less data = more variability =  greater spread.

Each t-value is different. Unlike the z-critical values, we don't have set values for 90%, 95%, and 99%. Therefore, we have to calculate the critical value based on the size of our sample.

To calculate a t* critical value, first draw the normal curve with your level of confidence in the middle. For instance, if I was using 95%, I would draw the curve and put 95% in the middle. Then, take whatever area is left over on the LOWER end of the curve only and add it to the middle. I would have 2.5% left over, so I would add it to 95% to get 97.5%. Then, on your TI-84, go to 2nd - distribution and go to #4: t-interval. Fill in your percentage, but put it as a decimal (.975) and put in your sample size. Hit calculate, and this gives you the t-critical value to use in your confidence interval formula!

The interpretation and calculation is the same for t-intervals. The only difference is that, under assumptions, we need to state that the sample size was not sufficiently large, so a t-distribution must be used.

Monday January 27th: Confidence Intervals with Proportions

Proportion confidence intervals are the same as x-bar confidence intervals, except we use p-hat and p instead. Use the exact same confidence interval formula:

p-hat +/- crit. val (std. dev) where the std. dev is equal to root(p-hat(1-p-hat)/n)

In most instances, you'll see something like "45 out of 56..." in a problem, so you'll need to use this information to calculate what your p-hat is. The procedure for a proportion confidence interval is the same except for your third assumption, you must check, in regard to sample size:

np-hat >= 10 and n(1 - p-hat) >=10. This verifies the size assumption, so by the Central Limit Theorem, the sample either is or is not sufficiently large.

Thursday January 23rd: Calculating Sample Size in a Confidence Interval

Let's suppose that you already have a confidence interval, but you want to know how many observations your sample needs in order for your interval to look that way. Or let's say that you want your interval to be a certain width: say, only 10 on both the plus and the minus side - what sample size will make sure that this is true? (The width of the confidence interval, by the way, is called a margin of error. It's what you get when you multiply the critical value by the standard deviation).

To find the sample size, set your margin of error equal to your critical value times your standard deviation:
MOE = (crit. val)(std. dev).
Divide each side by your critical value to cancel it out. Square each side to get rid of the radical. Cross multiply to get the n on the left of the equation so that it will be by itself, and divide the standard deviation to get your answer.

Wednesday January 22nd: More on Confidence Intervals

We know that in statistics all samples are different. I could take a sample of student GPAs, and Mrs. Cain could take a sample of student GPAs, and chances are that our mean GPA would not be exactly the same.

Since a confidence interval is created AROUND a sample parameter (x-bar or p-hat), and since all samples are different, all confidence intervals will then also be different. My 95% confidence interval will be different from Mrs. Cain's 95% confidence interval.

95% of the time, a 95% confidence interval will still capture the true population parameter. For example, if I took a sample and made a 95% confidence interval every day, 95% of the intervals that I make will have the true population mean in them. The other 5% won't - that's why we're only 95% confident that our interval is correct.

Tuesday January 21st: Confidence Intervals with Means

A confidence interval is used when we have a sample and want to estimate the corresponding population parameter. If we have x-bar, we want to estimate mu. If we have p-hat, we want to estimate p. We do this by following this formula (for either means or proportions):

sample +/- critical value(std. dev)

For means, this is x-bar +/- critical value (Sx/root(n))
For proportions, this is p-hat +/- critical value (root(p-pat(1-p-hat))/n).

A critical value is derived from the Normal Curve, and we use it to estimate the interval at a certain level of confidence. On the AP exam, you will typically be asked to calculate the interval using 90%, 95%, or 99% confidence, although we can certainly derive the other critical values using the Normal Curve.

The three critical values that you will need to know are:
90%: 1.645
95%: 1.96
99%: 2.58

When you construct your interval for x-bar, you must follow three important steps.
1) Check Assumptions. They are:
-Assume the data came from a simple random sample
-Assume that the data is independent: that the sample size is less than 10% of the population size
-Assume that the sample size is sufficiently large by the Central Limit Theorem. For x-bar, it must be greater than or equal to 30 observations to roughly resemble a Normal curve.
2) Calculate the interval by using
x-bar +/- critical value (Sx/root(n))
3) Interpret in Context:
"We are (percent) confident that the true population parameter (mu or p) for (whatever your problem is about in context) is between (the lower and upper part of your interval).

Hints for Problem Set 10

Problem 1:
(B) You need to give me an explanation using CORRECT statistical terminology to get full credit. Just writing "Central Limit Theorem," is going to earn you about a 2/10.....

Problem 2:
(A) You will have two x-bars for this question. This question is a bit tricky - my advice is to draw a picture of a Normal Curve and shade the area that you're trying to find. Once you find your answer, ask yourself, "Does this make sense?" You are finding the probability that it will shut down, or NOT work....

(B) I recommend finding what two standard deviations from the mean would be, first, then use those two values to find two separate z-scores. The difference of those two z-scores is your answer.

Problem 3:
This is a tough one because it doesn't explicitly tell you what the means are. Consider the population mean to be the total baggage limit divided by the number of passengers on the flight. It gives you the standard deviation, and suppose your sample mean is represented by the individual passenger.

EARLY CHRISTMAS PRESENT - Skip number four on the problem set. It is worded poorly. Instead of taking paragraphs on paragraphs to explain it, just don't worry about it - Happy Holidays.

Problem 5:
(A) Change p-hat to p. The mean is already given to you (think about what p=0.5 represents...) and standard deviation by doing the square root of p(1-p)/n

(C) Remember that you're looking for the upper part of the curve here, so think about what you would need to do to your probabilities to get the correct answer...

(D) Think about the Central Limit Theorem here, and the computer program that I showed you in class. What happens to the Normal Curve when there are more observations? And because of the way the Normal Curve looks, what then should happen to your probabilities?

Sampling Distributions with Proportions

Sampling distributions with proportions are very similar sampling distributions with means. We are still going to use the z-score formula, but we're going to use it with proportions parameters. They are:

p-hat, the sample proportion (instead of x-bar)
p, the population proportion (instead of mu)
std. dev. of p, which is found by taking the square root of p(1-p)/n
and n, the number of observations.

Use the same z-score formula (sample-population/std. dev) to solve.

I realize that this explanation is kind of vague, so please check the link below for a video example of how to solve this type of problem:

Central Limit Theorem

The Central Limit Theorem says that, with more observations, a distribution appears to look more and more like a Normal Curve (bell-shaped). With too few observations, the curve is too flat and we can't assume Normality, which means we cannot use Normalcdf.

The AP Stats rule is that, if the population standard deviation is known and if n>= 30 observations, then we can assume that the distribution takes a Normal Shape. If n is fewer than 30, then we can't use the Normal Distribution. We will have to use something called the t-distribution, which we won't go over until the beginning of next year.

Conceptual Stuff about Sampling Distributions

Here are a few key stats-y points about sampling distributions that you want to keep in mind, especially when you are doing your problem set (10 bonus points if you recommend me a good workout song at the end of your problem set!).

1. If your z-score is positive, then your x-bar should be greater than your mu. If your z-score is negative, then your x-bar should be lower than the mu. If they are the same, then the z-score should be zero and your probability will be 50%.

2. When n is low, the z-score becomes smaller. When n is higher, the z-score becomes larger (if all other variables are held constant).

3. Your z-score represents how many standard deviations your sample mean is from your population mean. If the z-score is within + or - 3 standard deviations from the mean, then we can assume that the sample is representative of the population without calculating the probability. If the z-score is more than -3 or +3 standard deviations from the mean, then are sample mean is too far from the population mean to say that the sample accurately represents the population.

4. If a problem asks you to find the probability that the sample is "between" two different sample means, find the probability of the first (Normalcdf(first z-score)), then find the probability of the second (Normalcdf(second z-score)), then subtract those probabilities.

Remember that a probability MUST be between zero and one - ALWAYS!!!

Intro to Sampling Distributions

A sampling distribution is a distribution based around a sample statistic, such as x-bar. We would much rather use the information from the population (mu) instead, but sometimes that information is not available or sometimes it is too hard/not possible to sample an entire population. So, we use a sampling distribution instead.

The main question then becomes: if we take a sample from the population, what is the probability that the sample statistic (x-bar) is actually representative of the population statistic (mu)? In other words, are the sample and population close enough to one another that they are essentially the same? That depends. To find out, we need a sample mean, a population mean, a population standard deviation, and n, which is the number of observations in the sample. Plug these values into the z-score formula

z = (x-bar - u)/(sigma/rad(n)). (Awkward writing the formula in the blog without all of the math symbols!)

This will give you the z-score. Then, plug this value into Normalcdf, and this will give you the probability that our sample represents the population. Note: if your x-bar is higher than mu, do 1-Normalcdf instead.

Hints for Problem Set 9

Problem 1: I am looking for a paragraph answer. Please be specific. If you choose to design an experiment, you may include an experiment diagram to SUPPORT your answer but that should not be your entire answer.

Problem 2: positively skewed means skewed to the right and negatively skewed means skewed to the left. You may use the TI to calculate your boxplot but you must include a scale and titles for full credit.

Problem 3: binomial/geometric probabilities, think about the stuff we just covered.

Problem 4: when you comment on the differences for both histograms that you made, with titles labels and a scale, use CUSS to do this most effectively, and comparator language.

Problem 5: this is probably the most challenging question. Note that you cannot describe linearity without a scatter plot. There are several key calculator functions that you'll need to do this problem - check your notes. Note that we determine transformations using improvements in both the residual plot and in r-squared. Don't forget to put the transformed variable in your new regression equation(ex y =3.2logx + 7)

If you need help for question 5, or any other question for that matter, please contact me directly or come after school tomorrow. Remember to always write in complete sentences and to never begin a sentence with "because"

Binomial Distribution to Normal Curve

Today we discussed instances where the binomial distribution could be transformed into the Normal Curve (because the Normal Curve is more accurate). This can happen if the number of trials (n) times the probability (p) is greater than 10. In other words:

np >= 10  for us to use the normal curve.

To solve, we must recall that the average of a binomial distribution is np, and the standard deviation of a normal distribution is the square root of np(1-p). If we're using the Normal curve, then we must use a z-score (because we use z-scores with Normal Curves). Remember that the formula for a z-score is:

z = x-bar - mu/std. dev.  

Calculate the mean (although you already should have, to check the assumption for Normal Curve) and the standard deviation, then plug it into the formula to get your z-score. Once you have your z-score, plug this into Normalcdf(z-score) and that will give you your probability!

Don't forget that we would do 1 - Normalcdf(z-score) if we wanted the top half of the curve.

The example that we did at the end of class today was: Suppose that there is a 5% chance that Mr. Guyton stops you in the hallway. If Amber surveys 300 students, what is the probability that less than 16 of them will have been stopped by Mr. Guyton?

For a video solution to this, please click here:

Binomial and Expected Value open response

We did two tough problems in class today! Well done for those of you who followed along and gave it your best shot.  VERY impressed with the efforts considering that it was the first day back from a holiday break.

For the second problem that we did, I realize that we didn't get to part D in class.  For those of you interested in increasing your statistical knowledge, here's a video solution to part D problem 2 (the one from 2005).

http://www.educreations.com/lesson/view/2005-part-d/14386947/?s=AHSFQB&ref=app

Problem Set 8 Hints

For better or for worse, I decided to do this round of hints as a series of videos. Check them out to get some information (and even some picture hints!) on how to solve these questions. There is a hint for every problem except number two, that one was pretty straightforward.

Don't forget to interpret in the context of the problem, use complete sentences, and bring your hot chips inspired dish for the Thanksgiving Iron Chef Competitin tomorrow!! +10 bonus points if you participate!!

http://www.educreations.com/lesson/view/probset8hints1/14020302/?s=aIbFX6&ref=app

http://www.educreations.com/lesson/view/probset8hints3/14020390/?s=MmLbqF&ref=app

http://www.educreations.com/lesson/view/probset8hints4/14020456/?s=9BEMj3&ref=app

http://www.educreations.com/lesson/view/probset8hints5/14020511/?s=8mHmXV&ref=app

Enjoy!

Binomial Distributions

Binomial distributions are discrete distributions where x, the random variable, can either succeed or fail at something. The problem will give you a probability (of success or failure), a number of trials, and how many successes they are looking for.  Your answer will be a probability and should be written in the form p(x = c) where c is the number that you are looking for in the problem.

We made binomial distributions in class today based off of tables and of graphs. Either one conveys the same information. I also discussed how to do this in the calculator. Go to 2nd - vars, then scroll down to binompdf. From there, type in the number of trials, the probability as a decimal between 0 and 1, and the number of successes.

Your answer should be interpreted as:"The binomial probability of getting (number of successes) given (number of trials) and (probability) of success is (your answer)."

Discrete and Uniform Distributions

A distribution is what the data looks like (to put it in very simplified terms). Distributions can either be discrete (exact values) or continuous (infinite decimal values). When we simulate a distribution, the center of the distribution/highest peak becomes known as the expected value of the distribution. This means that, given no other information, we'd expect a random trial to give us that value because it is the most common.

A uniform distribution is one where the probability of any value happening is the same throughout the distribution. Remember that the the probability under a curve (uniform or not) is always equal to 1. We did the following example of a uniform distribution in class on Friday but never quite got around to finishing it. Click on the link below to get the whole video solution to the following question:

"Let x be the time in minutes that a commuter must wait for a public transit train, with the minimum wait time being 0 minutes and the maximum wait time being 20 minutes. Suppose that the writ times are uniformly distributed.
A. Draw a density curve to model the distribution.
B. what is the probability that x is less than 10 minutes?
C.  What is the probability that x is between 7 and 12 minutes?
D.  Find the value of c for which P(x < c) = .9.

http://www.educreations.com/lesson/view/uniform-distribution/13711048/?s=LDKaL9&ref=app

Simulations

We have studied simulations over the past few days - once in the context of goldfish and once in the context of trying to recall as many 3-letter words as possible within a 1-minute time period. Simulations are repeated experiments. We perform simulations to gain insight on what the population distribution looks like. The more simulations we perform, the less variability we have, and the more the sample distribution looks like the population distribution.

When describing a simulation, we always:
1. State the model
2. Define what one trial is, and mention if there is a stopping rule.  Note that the AP exam requires at least 100 trials for a simulation to be large enough
3.  State what is being recorded (I.e. what is the random variable?)
4.  Perform the simulation and collect results
5.  Analyze the results