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).
Sunday, February 2, 2014
Tuesday, December 17, 2013
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?
(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?
Monday, December 16, 2013
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:
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:
Saturday, December 14, 2013
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.
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!!!
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.
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.
Monday, December 9, 2013
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"
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"
Subscribe to:
Posts (Atom)