AP Statistics Page 1 of 4 Unit Overview: Binomials and Distributions Think about It

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The following questions will help you study key concepts covered in this unit. 1. You're taking samples of size 10 from a population of 20 marbles and recording the

number of blue marbles in each sample. Why isn't this a binomial situation, and why would it be almost binomial if you did the same experiment with a population of 200 marbles?

2. Explain the term continuity correction. Why does it work, and how do you use it?

3. Consider a parent population with mean 75 and standard deviation 7. The population

doesn't appear to have extreme skewness or outliers.

A. What are the mean and standard deviation of the distribution of sample means for n = 40?

B. What's the shape of the distribution? Explain your answer in terms of the central

limit theorem. C. What proportion of the sample means of size 40 would you expect to be 77 or less?

If you use your calculator, show what you entered. D. Draw a sketch of the probability you found in part C, and label the horizontal axis.

4. Suppose that after several years of offering AP Statistics, a high school finds that final

exam scores are normally distributed with mean 78 and standard deviation 6.

A. What are the mean, standard deviation, and shape of the distribution of x-bar for n = 50?

B. What's the probability a sample of scores will have a mean greater than 80? C. Sketch the distribution curve for part B, showing the area that represents the

probability you found. Be sure to label the horizontal axis. 5. In a survey, 600 mothers and fathers were asked about the importance of sports for

boys and girls. Of the parents interviewed, 70% said the genders are equal and should have equal opportunities to participate in sports.

A. What are the mean, standard deviation, and shape of the distribution of the sample

proportion p of parents who say the genders are equal and should have equal opportunities? Be sure to justify your answer for the shape of the distribution. Use n = 600.

B. Using the normal approximation without the continuity correction, sketch the probability distribution curve for the distribution of p. Shade equal areas on both sides of the mean to show an area that represents a probability of .95, and label the upper and lower bounds of the shaded area as values of p-hat (not z-scores).

C. In your sketch from part B, the shaded area shows a .95 probability of what

happening? In other words, what does the probability of .95 represent?

AP Statistics Page 2 of 4 Unit Overview: Binomials and Distributions Think About It

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D. Using the normal approximation, what's the probability that a randomly drawn sample of 600 parents will have a sample proportion between 67% and 73%? Draw a sketch of the probability curve, shade the area representing the probability you're finding, and label the z-scores that represent the upper and lower bounds of the probability you're finding. Don't use the continuity correction.

E. Now, use the exact binomial calculation to find the probability of getting between,

but not including, 67% and 73% of the respondents in a sample of 600 who say the genders are equal and should have equal opportunities. To use the exact binomial, you'll need to convert the proportions to counts by multiplying each proportion by 600.

F. Now try it again, but this time find the probability of getting at least 67% but no

more than 73%. Use the exact binomial calculation. 6. Annie is a basketball player who makes, on average, 65% of her free throws. Assume

each shot is independent and the probability of making any given shot is .65.

A. What's the probability Annie will miss three straight free throws before she makes one? (If you use a calculator to get your answer, write your answer in standard notation and show what you do on the calculator as well.)

B. During a season, Annie takes 140 free throws. What's the exact binomial probability

she'll make at least 100 out of 140 of these throws?

C. For part B, are the conditions that permit you to use a normal approximation to the binomial satisfied? Explain.

D. Redo part B using a normal approximation, without continuity correction, to the

binomial. Draw a sketch of the situation. Then draw the distribution, shade the area representing the probability you're finding, and label the horizontal

axis approximately. NOTE: If you use a graphing calculator, show what you do to get the z-scores, and explain your answer in enough detail to show your instructor you understand what you're doing.

E. Redo part B using a normal approximation, with continuity correction, to the

binomial. NOTE: If you use a graphing calculator, show what you do to get the z-scores on your calculator, and explain your answer in enough detail to show your instructor you understand what you're doing.

AP Statistics Page 3 of 4 Unit Overview: Binomials and Distributions Think About It

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Discussion 7. Have you ever been stuck in Jail when playing Monopoly? To get out, you need to roll doubles in

three tries or fewer or you have to pay. On average, how many times would people have to roll before getting doubles, and is that number larger than three? This is an average waiting-time question, where you're interested in the number of tries you need on average to get the outcome you want. You'll learn more about situations like this and the probability distributions they produce in the next Tutorial. Meanwhile, this Discussion will help you think about waiting-time situations. Respond to any one (or more) of the following, or respond to another student's posting. As you explain your reasoning, be sure to use what you know about the laws of probability.

1. If you buy a very large bag of candies colored brown, yellow, green, blue, orange, and red, and you start eating them, how many candies would you expect to pick until you got a blue one? Explain your reasoning using what you know about probability.

2. What's the probability you'll get out of Jail in Monopoly without having to pay? In other words, what are the chances you'll roll doubles on two six-sided dice within three tries?

3. You may be familiar with promotional campaigns where a company's products are marked with a letter, under the bottle cap of a soft drink, for example, and you're supposed to spell something to win a prize. In these cases, do you think some game pieces are more common than others? Describe an experiment you could conduct to see if some game pieces are easier to get than others.

4. Say you're playing a game like the one described in topic 3. A soft drink company has a letter printed on each bottle cap and the object is to spell the words I bought a lot of bottles to spell this. You have all the letters you need except p. The company's disclaimer statement says for each bottle you buy there's a 1/200 chance of getting a p. On average, how many bottles would you expect most people to buy in order to get this letter?

5. Create your own question about an average = waiting-time situation, or describe a real one you've seen or participated in. Explain why it's an average = waiting-time situation, and invite other students to answer it.

6. Have you ever won anything in a game like the ones described in topics 3 and 4? Describe the game, and calculate the probability of winning.

8. A sampling distribution is the distribution of a statistic. In other words, if you took the mean of many samples from a population, the set of means would form the distribution of . Some of the questions below have important information you'll need in order to understand the other questions, so before choosing which question to answer please read them all. Respond to one or more of the following or respond to another student's posting.

1. Suppose you had a population of fish whose lengths were normally distributed with a mean of 50 centimeters and a standard deviation of 5 centimeters. You draw a simple random sample of size 10, record the length of each fish, and calculate the mean of the sample lengths. What do you think your sample mean would most likely be? Explain your answer using what you know about random samples and sample means.

AP Statistics Page 4 of 4 Unit Overview: Binomials and Distributions Think About It

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2. For the scenario in topic 1, imagine you draw hundreds of samples of size 10 (replacing each fish after you record its length) and calculate the mean of each sample. If you kept doing this until you took every possible sample, you'd get a distribution of sample means called a sampling distribution. What would the shape of the sampling distribution be? Would the standard deviation of the sampling distribution be larger or smaller than the population standard deviation? Explain your answer using what you know about random samples and sample standard deviations.

3. Why might a sampling distribution be helpful if you wanted to estimate the likelihood of getting a particular sample value, say x-bar = 48 centimeters? What's your guess about the likelihood of getting a sample mean of 48 centimeters? Explain your answer using what you know about probability distributions.

4. Building on items 1-3, how might a sampling distribution be helpful if you wanted to estimate a population mean with a sample mean? Outline a scenario for using a sampling distribution for inference.

5. Write a paragraph-long story about a sampling distribution; describe the population, the sampling method, the mean and standard deviation of the population and the sampling distribution.

6. There's a case where a sampling distribution qualifies as a binomial setting. Describe this case, using the criteria for a binomial distribution.