Problem Solver
Instead of "Practice Problems" I have decided to provide a decision tree/map to help you identify the type of problem you have, and then solve an example that will hopefully help you with problems you are working on.
There may be several ways to arrive at the same thing, so just proceed through the questions and see if it gets you to where you want to be. As always, don't forget to search the Q&A page as well as the Statistics Topics page. If the following doesn't help you, just send me your question on the Contact page.
In Each Case, click on the answer to take you to the next question in the decision process. Then there will be an example problem and another click will take you to the answer.
(Please be patient - this is a work in progress! Feel free to offer some suggestions.)
What kind of Problem Is it?
- Find A Probability
- Perform Hypothesis Test
- Calculate a Confidence Interval
- Run/Interpret Regression or ANOVA
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Find a Probability - What Kind of Variable is it?
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Perform Hypothesis Test - What Kind?
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Calculate a Confidence Interval - For what?
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Run/Interpret Regression or ANOVA - (coming soon...)
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Find a Probability: Single Normal Variable
Example: "X is Normal with mean 100 and STD 20. Find the probability that X is between 70 and 110" (answer)
Example: "X is Normal with mean 100 and STD 20. Find the 80th percentile (or find the value that only 20% of the distribution is larger than" (answer)
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Find a Probability: The Mean of a Sample
Example: "A sample of 25 is taken from a population that has a mean of 100 and a STD of 20. Find the probability that the sample mean will be between 90 and 104" (answer)
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Find a Probability: A Binomial
Example: "There are 10 multiple choice questions, each with 4 answers. Therefore the probability of guessing a correct answer on any one of the questions is .25. If someone guesses on all 10, what is the probability they get exactly 4 correct? What is the probability of AT LEAST 4 right?" (answer)
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Find a Probability: A Sample Proportion
Example: "50 Students across the country are sampled and asked whether they smoke (Yes/No). Since it is known that 35% of all students smoke, what is the probability that more than 20 of those 50 will say they smoke?" (answer)
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Perform Hypothesis Test: For a single mean
Example: "There is a claim that the mean width of widgets produced at factory A are 10 mm. A sample of 25 widgets produces a mean width of 11.4 mm. Test the factory's claim, assuming the Std Dev is 2.0mm" (answer)
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Perform Hypothesis Test: For two means
Example: "Two groups of students are given the SAT. The first group is given tutoring before the exam, the second group is not. Test the hypothesis that the tutoring improves the average SAT score. The relevant information is as follows: Group1: n=20, mean SAT = 1130 (SD = 50); Group2: n=25, mean SAT = 1090 (SD = 45)." (answer)
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Perform Hypothesis Test: For a single proportion
Ex
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Perform Hypothesis Test: For two proportions
Ex
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Perform Hypothesis Test: For a Paired Difference
Ex
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Calculate a Confidence Interval: For a population mean from a sample mean
Example: "A sample of 25 widgets from factory A has a mean width of 11.4 mm. Assuming the Std Dev is 2.0 mm, create a 95% confidence interval for the mean width of widgets produced from factory A." (answer)
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Calculate a Confidence Interval: For a population proportion from a sample proportion
Ex
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Example: "X is Normal with mean 100 and STD 20. Find the probability that X is between 70 and 110"
This is pretty straight forward: P(70 < X < 110) = P([70-100]/20 < Z < [110 – 100]/20 ) = P(-1.5 < Z < 0.5) which we can find the following way:
P(Z < 0.5) – P(-1.5) = 0.69 - .07 = 0.62
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Example: "X is Normal with mean 100 and STD 20. Find the 80th percentile (or find the value that only 20% of the distribution is larger than"
These are both asking the same question. Find the value in the formula below that makes it true:
P(X < ___ ) = .8
We proceed as we usually do, by standardizing X:
P(Z < [___ - 100]/20) = .8
We then have to find the z-score that makes that true. I get a z-score of .84. That is:
P(Z < .84) = .8, so we know that the quantity [___ - 100] / 20 must equal .84. Set equal to each other ans solve, giving: ___ = (.84)*20 + 100 = 116.8
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Example: "A sample of 25 is taken from a population that has a mean of 100 and a STD of 20. Find the probability that the sample mean will be between 90 and 104"
Since X is Normal we know that the sample mean (xbar) is too. (If X wasn’t Normal, a large enough sample size makes xbar Normal anyway.) xbar is Normal with mean 100 and STD equal to the entire quantity: 20/sqrt{25}, or 20/5 = 4.0. So now we proceed as usual:
P(90 < xbar < 104) = P([90 – 100]/4.0 < Z < [104-100]/4.0) = P(-2.5 < Z < 1.0) which we can solve as:
P(Z < 1.0) – P(Z < -2.5) = 0.841 - .006 = 0.835
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Example: "There are 10 multiple choice questions, each with 4 answers. Therefore the probability of guessing a correct answer on any one of the questions is .25. If someone guesses on all 10, what is the probability they get exactly 4 correct? What is the probability of AT LEAST 4 right?"
X is Binomial in this case, so we can find individual probabilities from the binomial formula. For instance:
P(X = 4) = (10 choose 4) * .25^4* .75^6 = 0.146. To find the probability of “at least 4”, we would have to either calculate P(X=4) + P(X=5) + … + P(X=10), or we could find P(X=0)+P(X=1)+P(X=2)+P(X=3) and subtract it from 1. The answer is .224
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Example: "50 Students across the country are sampled and asked whether they smoke (Yes/No). Since it is known that 35% of all students smoke, what is the probability that more than 20 of those 50 will say they smoke?"
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Example: Hypothesis Test for two means
X
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