Statistics 1.1, 1.2, and 1.3

How to determine standard deviation.

p*(1-p)/n

Suppose a friend of yours says she is a 75% free-throw shooter in basketball. You don't think she is that good and want to test her to gather evidence that she makes less than 75% of her free throws in the long run. You have her shoot 40 free throws and she makes 26 (or 65%) of them. Based on this information, you make the following null distribution. What is the value of the standardized statistic for your friend? -1.47 -0.13 -0.10 1.47 4.77

-1.47

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 12 out of 20, or 60%, of his serves in. Jerry wants to know if his long-run proportion of getting his first serve in is greater than 0.50. What values would you enter into the One Proportion applet to run an analysis for Jerry? Probability of heads:

0.50

Tennis players often spin a racquet to decide who serves first. The spun racquet can land with the manufacturer's label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down. What value does the chance model assert for the long-run proportion?

0.50

Many types of studies have been done to see whether animals have a sense of number. In one study (Beran & Beran, 2004), researchers wanted to see whether chimpanzees could remember how many items were added to a container when they see them added one at a time. To do this they had two containers set up so the chimpanzees could not see inside them. They randomly added bananas to the containers one at a time and then had the chimpanzees choose the container they wanted. In one set of experiments, they added 3 bananas to one container and 4 to the other and had a chimpanzee named Mercury pick which container he wanted. In 20 trials, he picked the container containing 4 bananas 16 times (or 80%). Do these data convince you that Mercury is more likely to pick the container with more bananas in it in the long run? A null distribution, with 100 repetitions, for testing this situation is shown. Using the null distribution shown, what is the smallest proportion of times that Mercury could pick the container containing 4 bananas where you would say there is strong evidence that Mercury is more likely to pick the container with more bananas in it in the long run?

0.75

A chimpanzee named Sarah was the subject in a study of whether chimpanzees can solve problems. Sarah was shown 30-second videos of a human actor struggling with one of several problems (for example, not able to reach bananas hanging from the ceiling). Then Sarah was shown two photographs, one that depicted a solution to the problem (like stepping onto a box) and one that did not match that scenario. Researchers watched Sarah select one of the photos, and they kept track of whether Sarah chose the correct photo depicting a solution to the problem. Sarah chose the correct photo in seven of eight scenarios that she was presented. What is the observed value of the statistic in this case?

0.875

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 60 out of 100, or 60%, of his serves in. Jerry wants to know whether his long-run proportion of getting his first serve in is greater than 0.50. Use the One Proportion applet to test if Jerry's long-run proportion of getting his first serve in is greater than 0.50. State the values you would enter into the applet. Number of tosses:

100

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 12 out of 20, or 60%, of his serves in. Jerry wants to know if his long-run proportion of getting his first serve in is greater than 0.50. What values would you enter into the One Proportion applet to run an analysis for Jerry? Number of repetitions:

1,000

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 60 out of 100, or 60%, of his serves in. Jerry wants to know whether his long-run proportion of getting his first serve in is greater than 0.50. Use the One Proportion applet to test if Jerry's long-run proportion of getting his first serve in is greater than 0.50. State the values you would enter into the applet. Number of repititions:

1,000 or more

If you drop a piece of buttered toast on the floor, is it just as likely to land buttered side up as buttered side down? It sure seems like mine always lands buttered side down! Suppose that 7 of the last 10 times I dropped toast it landed buttered side down. In order to carry out a statistical analysis, the One Proportion applet was used with 100 repetitions to see if my toast fell buttered side down a majority (more than 50%) of the time. How many dots are in the dotplot?

100

One of the authors sometimes likes to play Minesweeper, and of the last 20 times she played Minesweeper, she won 12 times. That is, she won 60% of the games.​The One Proportion applet was used to generate 100 possible values. The dotplot generated below by the applet assumed the long-run proportion of winning was equal to 0.50. How many dots are in the dotplot?

100

Although we know some animals, like dogs, have a keen sense of smell, what about humans? A woman from Scotland, Joy Milne, claimed that she was able to determine whether someone had Parkinson's disease just based on their smell (Morgan, 2016). Her husband died from the disease and she noticed that he started smelling different prior to his diagnosis. To test this, researchers had six people known to have Parkinson's disease and six people thought not to have the disease wear t-shirts for a day. They collected the shirts and then tested Milne. She correctly identified 11 of the 12 shirts. Suppose we let heads represent a correct guess and tails represent an incorrect guess. We would need to flip a coin _____ times to model outcomes from someone just guessing whether a shirt came from a Parkinson's patient on the test the researchers gave Joy Milne.

12

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 12 out of 20, or 60%, of his serves in. Jerry wants to know if his long-run proportion of getting his first serve in is greater than 0.50. What values would you enter into the One Proportion applet to run an analysis for Jerry? Number of tosses:

20

If you drop a piece of buttered toast on the floor, is it just as likely to land buttered side up as buttered side down? It sure seems like mine always lands buttered side down! Suppose that 7 of the last 10 times I dropped toast it landed buttered side down. In order to carry out a statistical analysis, the One Proportion applet was used with 100 repetitions to see if my toast fell buttered side down a majority (more than 50%) of the time. At what number will the resulting dotplot be centered?

5

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 60 out of 100, or 60%, of his serves in. Jerry wants to know whether his long-run proportion of getting his first serve in is greater than 0.50. Use the One Proportion applet to test if Jerry's long-run proportion of getting his first serve in is greater than 0.50. State the values you would enter into the applet. Probability of heads:

50

Variable

A factor that can change in an experiment

Suppose a friend of yours says she is a 75% free-throw shooter in basketball. You don't think she is that good and want to test her to gather evidence that she makes less than 75% of her free throws in the long run. You have her shoot 40 free throws and she makes 26 (or 65%) of them. You run this test and find a p-value of 0.1150. Which of the following is the best way to state the conclusion? -Because the p-value is large, there is strong evidence that your friend is a 75% free-throw shooter in the long run. -Because the p-value is small, there is strong evidence that your friend is a 75% free-throw shooter in the long run. -Because the p-value is small, there is strong evidence that your friend is less than a 75% free-throw shooter in the long run. -Because the p-value is not small enough, there is not strong evidence that your friend is less than a 75% free-throw shooter in the long run.

Because the p-value is NOT small enough, there is NOT strong evidence that your friend is less than a 75% free-throw shooter in the long run.

Although we know some animals, like dogs, have a keen sense of smell, what about humans? A woman from Scotland, Joy Milne, claimed that she was able to determine whether someone had Parkinson's disease just based on their smell (Morgan, 2016). Her husband died from the disease and she noticed that he started smelling different prior to his diagnosis. To test this, researchers had six people known to have Parkinson's disease and six people thought not to have the disease wear t-shirts for a day. They collected the shirts and then tested Milne. She correctly identified 11 of the 12 shirts. How is your answer from the previous part related to the new location of the statistic in the simulated distribution in the applet?

Compared to 11, 12 is farther away in the tail and farther away from 6 which is the center of distribution of correct guesses.

One of the authors sometimes likes to play Minesweeper, and of the last 20 times she played Minesweeper, she won 12 times. That is, she won 60% of the games.​The One Proportion applet was used to generate 100 possible values. The dotplot generated below by the applet assumed the long-run proportion of winning was equal to 0.50. What does each dot represent in terms of Minesweeper games and wins?

Each dot represents the number of times out of 1,000 attempts the author wins a game of Minesweeper when the probability that the author wins is 0.50.

Suppose a friend of yours says she is a 75% free-throw shooter in basketball. You don't think she is that good and want to test her to gather evidence that she makes less than 75% of her free throws in the long run. You have her shoot 40 free throws and she makes 26 (or 65%) of them. Set up the correct null and alternative hypotheses for this test. Ho- Ha-

Ho- (pi = 0.75) Ha- (pi < 0.75)

Many types of studies have been done to see whether animals have a sense of number. In one study (Beran & Beran, 2004), researchers wanted to see whether chimpanzees could remember how many items were added to a container when they see them added one at a time. To do this they had two containers set up so the chimpanzees could not see inside them. They randomly added bananas to the containers one at a time and then had the chimpanzees choose the container they wanted. In one set of experiments, they added 3 bananas to one container and 4 to the other and had a chimpanzee named Mercury pick which container he wanted. In 20 trials, he picked the container containing 4 bananas 16 times (or 80%). Do these data convince you that Mercury is more likely to pick the container with more bananas in it in the long run? A null distribution, with 100 repititions, for testing this situation is shown. Let π be the long-run proportion of times Mercury picks the container with more bananas. Set up the correct null and alternative hypotheses for this study in symbols. Ho Ha

Ho= (pi = 0.50) Ha= (pi > 0.50)

A chimpanzee named Sarah was the subject in a study of whether chimpanzees can solve problems. Sarah was shown 30-second videos of a human actor struggling with one of several problems (for example, not able to reach bananas hanging from the ceiling). Then Sarah was shown two photographs, one that depicted a solution to the problem (like stepping onto a box) and one that did not match that scenario. Researchers watched Sarah select one of the photos, and they kept track of whether Sarah chose the correct photo depicting a solution to the problem. Sarah chose the correct photo in seven of eight scenarios that she was presented. Which of the following best describes the meaning of the p-value in this situation? --If Sarah understands how to solve problems and is not just guessing at which picture to select, the probability she would get 5 or more correct out of 8 is 0.04. --If Sarah doesn't understand how to solve problems and is just guessing at which picture to select, the probability she would get 7 or more correct out of 8 is 0.04. --If Sarah understands how to solve problems and is not just guessing at which picture to select, the probability she would get 7 or more correct out of 8 is 0.04. --If Sarah doesn't understand how to solve problems and is randomly choosing which picture to select, the probability she would get less than 7 correct out of 8 is 0.04.

If Sarah doesn't understand how to solve problems and is just guessing at which picture to select, the probability she would get 7 or more correct out of 8 is 0.04.

Tennis players often spin a racquet to decide who serves first. The spun racquet can land with the manufacturer's label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down. Suppose that the spun racquet lands with the label facing up 48 times out of 100. What does this result suggest about the fairness of the spinning process?

It is plausible that the spinning process is not fair.

If you drop a piece of buttered toast on the floor, is it just as likely to land buttered side up as buttered side down? It sure seems like mine always lands buttered side down! Suppose that 7 of the last 10 times I dropped toast it landed buttered side down. In order to carry out a statistical analysis, the One Proportion applet was used with 100 repetitions to see if my toast fell buttered side down a majority (more than 50%) of the time. Based on the dotplot, does this prove that my long-run proportion of dropping toast buttered side down is 0.50?

No, 0.50 is just a plausible value of the parameter.

One of the authors sometimes likes to play Minesweeper, and of the last 20 times she played Minesweeper, she won 12 times. That is, she won 60% of the games.​The One Proportion applet was used to generate 100 possible values. The dotplot generated below by the applet assumed the long-run proportion of winning was equal to 0.50. Based on 12 wins on the last 20 games and the above dotplot, is there evidence that the author's long-run proportion of winning at Minesweeper is greater than 50%? Why or why not?

No, 12 is the most typical outcome since the long-run probability is 0.60.

One of the authors sometimes likes to play Minesweeper, and of the last 20 times she played Minesweeper, she won 12 times. That is, she won 60% of the games.​The One Proportion applet was used to generate 100 possible values. The dotplot generated below by the applet assumed the long-run proportion of winning was equal to 0.50. Based on 12 wins on the last 20 games and the above dotplot, there is not evidence in favor of the claim that the author's long-run proportion of winning at Minesweeper is higher than 50%. Does this prove that the author's long-run proportion of winning is equal to 50%?

No, 50% is just a plausible (reasonable) explanation for the data.

If you drop a piece of buttered toast on the floor, is it just as likely to land buttered side up as buttered side down? It sure seems like mine always lands buttered side down! Suppose that 7 of the last 10 times I dropped toast it landed buttered side down. In order to carry out a statistical analysis, the One Proportion applet was used with 100 repetitions to see if my toast fell buttered side down a majority (more than 50%) of the time. Based on 7 out of 10 slices of toast landing buttered side down and the dotplot, would you be convinced that the long-run proportion of times my toast lands buttered side down is greater than 50%, and why?

No, 7 is fairly typical outcome assuming the parameter is 0.50.

Tennis players often spin a racquet to decide who serves first. The spun racquet can land with the manufacturer's label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down. Suppose that the spun racquet lands with the label facing up 48 times out of 100. Does this result constitute strong evidence against believing that the spinning process is fair? Why or why not?

No, getting 48 out of 100 face up is a typical result, thus providing the process is fair.

A p-value is calculated assuming that which hypothesis is true: null or alternative?

Null hypothesis

A chimpanzee named Sarah was the subject in a study of whether chimpanzees can solve problems. Sarah was shown 30-second videos of a human actor struggling with one of several problems (for example, not able to reach bananas hanging from the ceiling). Then Sarah was shown two photographs, one that depicted a solution to the problem (like stepping onto a box) and one that did not match that scenario. Researchers watched Sarah select one of the photos, and they kept track of whether Sarah chose the correct photo depicting a solution to the problem. Sarah chose the correct photo in seven of eight scenarios that she was presented. What is the observed value of the statistic in this case? --Null: The long-run proportion of times Sarah chooses the correct photo is less than 0.50; H0: π < 0.50.​Alt: The long-run proportion of times Sarah chooses the correct photo is more than 0.50; Ha: π > 0.50. --Null: The long-run proportion of times Sarah chooses the incorrect photo is 0.50; H0: π = 0.50.​Alt: The long-run proportion of times Sarah chooses the incorrect photo is more than 0.50; Ha: π > 0.50. --Null: The long-run proportion of times Sarah chooses the correct photo is 0.50; H0: π = 0.50.​Alt: The long-run proportion of times Sarah chooses the correct photo is more than 0.50; Ha: π > 0.50. --Null: The long-run proportion of times Sarah chooses the correct photo is less than 0.50; H0: π < 0.50.​Alt: The long-run proportion of times Sarah chooses the correct photo is 0.50; Ha: π = 0.50.

Null: the long-run proportion of times Sarah chooses the correct photo is less than 0.50; Ha: (pi=0.50)

Tennis players often spin a racquet to decide who serves first. The spun racquet can land with the manufacturer's label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down. What statistical term is given to the true proportion of times the racquet lands face up?

Parameter

One of the authors sometimes likes to play Minesweeper, and of the last 20 times she played Minesweeper, she won 12 times. That is, she won 60% of the games.​The One Proportion applet was used to generate 100 possible values. The dotplot generated below by the applet assumed the long-run proportion of winning was equal to 0.50. At what number is the dotplot centered?

Since the chance model assumes a fair coin, the dotplot is centered at 0.5 times 20, which is 10.

Although we know some animals, like dogs, have a keen sense of smell, what about humans? A woman from Scotland, Joy Milne, claimed that she was able to determine whether someone had Parkinson's disease just based on their smell (Morgan, 2016). Her husband died from the disease and she noticed that he started smelling different prior to his diagnosis. To test this, researchers had six people known to have Parkinson's disease and six people thought not to have the disease wear t-shirts for a day. They collected the shirts and then tested Milne. She correctly identified 11 of the 12 shirts. Milne was adamant that one of the shirts came from someone with the disease when, in fact, that person had not been diagnosed. However, eight months later, the person Milne claimed had Parkinson's was given a diagnosis that he had the disease. So in fact Milne got all 12 correct! Does this now give stronger or weaker evidence that she is doing better than just guessing compared to what you thought when answering the previous part?

Strong evidence, because 12 correct is likely.

Suppose you are conducting a test of significance to try to determine whether your cat, Hope, will go to the correct object (out of two) when it is pointed to, just like Harley the dog did in the Exploration 1.1. You test Hope 100 times and finds she goes to the correct object in 70 of those 100 trials. In this scenario, what are the observational units?

The 100 trials

Observational Unit

The amount of trials

Which of the following statements is true if an observed statistic from a study turns out to be a likely value under the chance model? --The chance model is true. --There is evidence against the chance model. --The chance model is plausible. --None of the choices are correct because this is not real data.

The chance model is plausible

Parameter

The long-term probability of picking the correct object

Suppose you are conducting a test of significance to try to determine whether your cat, Hope, will go to the correct object (out of two) when it is pointed to, just like Harley the dog did in the Exploration 1.1. You test Hope 100 times and finds she goes to the correct object in 70 of those 100 trials. In this scenario, what is the parameter?

The long-term proportion (probability) that Hope will go to the correct object

A chimpanzee named Sarah was the subject in a study of whether chimpanzees can solve problems. Sarah was shown 30-second videos of a human actor struggling with one of several problems (for example, not able to reach bananas hanging from the ceiling). Then Sarah was shown two photographs, one that depicted a solution to the problem (like stepping onto a box) and one that did not match that scenario. Researchers watched Sarah select one of the photos, and they kept track of whether Sarah chose the correct photo depicting a solution to the problem. Sarah chose the correct photo in seven of eight scenarios that she was presented. What does a single dot in the null distribution represent in terms of Sarah and the photos?

The number of times Sarah would choose the correct picture (out of 8) if she were just guessing.

If you drop a piece of buttered toast on the floor, is it just as likely to land buttered side up as buttered side down? It sure seems like mine always lands buttered side down! Suppose that 7 of the last 10 times I dropped toast it landed buttered side down. In order to carry out a statistical analysis, the One Proportion applet was used with 100 repetitions to see if my toast fell buttered side down a majority (more than 50%) of the time. What does each dot represent in terms of dropped toast and buttered side down?

The number of times out of 10 attempts the toast lands buttered side down when the probability that the toast lands buttered side down is 50%

Suppose you are conducting a test of significance to try to determine whether your cat, Hope, will go to the correct object (out of two) when it is pointed to, just like Harley the dog did in the Exploration 1.1. You test Hope 100 times and finds she goes to the correct object in 70 of those 100 trials. In this scenario, what is the statistic?

The proportion of the 100 trials that Hope goes to the correct object

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 60 out of 100, or 60%, of his serves in. Jerry wants to know whether his long-run proportion of getting his first serve in is greater than 0.50. Use the One Proportion applet to test if Jerry's long-run proportion of getting his first serve in is greater than 0.50. State the values you would enter into the applet. Suppose that Jerry had stopped at 20 serves, having gotten 12 in. Even though the sample statistic is 60% in both cases, why do you suppose it is possible that you make a different conclusion in part (d) of this question using the 20 serves?

The sample size was different (20 serves vs 100 serves).

Tennis players often spin a racquet to decide who serves first. The spun racquet can land with the manufacturer's label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down.​Suppose that the spun racquet lands with the label facing up 24 times out of 100.​ What does this result suggest about the fairness of the spinning process?

There is statistically significant evidence that the pinning process is not fair.

One of the authors sometimes likes to play Minesweeper, and of the last 20 times she played Minesweeper, she won 12 times. That is, she won 60% of the games.​The One Proportion applet was used to generate 100 possible values. The dotplot generated below by the applet assumed the long-run proportion of winning was equal to 0.50. Suppose that it so happened that when the author played the last 20 games, she was also watching her favorite TV show. Which of the following are appropriate conclusions based on this information?

There were special circumstances when the author played these 20 games and so these 20 games may not be a good representation of the author's long-run proportion of wins in Minesweeper.

Many types of studies have been done to see whether animals have a sense of number. In one study (Beran & Beran, 2004), researchers wanted to see whether chimpanzees could remember how many items were added to a container when they see them added one at a time. To do this they had two containers set up so the chimpanzees could not see inside them. They randomly added bananas to the containers one at a time and then had the chimpanzees choose the container they wanted. In one set of experiments, they added 3 bananas to one container and 4 to the other and had a chimpanzee named Mercury pick which container he wanted. In 20 trials, he picked the container containing 4 bananas 16 times (or 80%). Do these data convince you that Mercury is more likely to pick the container with more bananas in it in the long run? A null distribution, with 100 repititions, for testing this situation is shown. Would you say that we have strong evidence that Mercury is not just randomly picking a container but has a sense of which container contains more bananas? Choose the best among the statements below.

We have strong evidence that Mercury is not randomly picking a container because 16 out 20 (0.80) rarely occurs by chance (if Mercury is just guessing) leading to a small p-value.

A chimpanzee named Sarah was the subject in a study of whether chimpanzees can solve problems. Sarah was shown 30-second videos of a human actor struggling with one of several problems (for example, not able to reach bananas hanging from the ceiling). Then Sarah was shown two photographs, one that depicted a solution to the problem (like stepping onto a box) and one that did not match that scenario. Researchers watched Sarah select one of the photos, and they kept track of whether Sarah chose the correct photo depicting a solution to the problem. Sarah chose the correct photo in seven of eight scenarios that she was presented. Based on the p-value, what can be said about Sarah's ability to understand how to solve problems similar to those she was presented?

We have strong evidence that Sarah is not simply guessing, because 7 out of 8 rarely occur by chance (if just guessing)

Tennis players often spin a racquet to decide who serves first. The spun racquet can land with the manufacturer's label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down.​Suppose that the spun racquet lands with the label facing up 24 times out of 100.​ Does this result constitute strong evidence against believing that the spinning process is fair? Why or why not?

Yes, if the spinning process was fair (50% chance of the racquet landing face up), getting 24 out of 100 face up is an atypical result.

Although we know some animals, like dogs, have a keen sense of smell, what about humans? A woman from Scotland, Joy Milne, claimed that she was able to determine whether someone had Parkinson's disease just based on their smell (Morgan, 2016). Her husband died from the disease and she noticed that he started smelling different prior to his diagnosis. To test this, researchers had six people known to have Parkinson's disease and six people thought not to have the disease wear t-shirts for a day. They collected the shirts and then tested Milne. She correctly identified 11 of the 12 shirts. Use the One Proportion applet to answer the question, "Is it unlikely that someone could identify at least 11 out of 12 shirts correctly if they were just guessing?" Which of the following is the best answer based on the results in the applet?

Yes, it is unlikley because at least 11 correct (heads) happened very rarely on 12 coin flips.

Although we know some animals, like dogs, have a keen sense of smell, what about humans? A woman from Scotland, Joy Milne, claimed that she was able to determine whether someone had Parkinson's disease just based on their smell (Morgan, 2016). Her husband died from the disease and she noticed that he started smelling different prior to his diagnosis. To test this, researchers had six people known to have Parkinson's disease and six people thought not to have the disease wear t-shirts for a day. They collected the shirts and then tested Milne. She correctly identified 11 of the 12 shirts. Would you say that we have strong evidence that Joy Milne tends to do better than just guessing in picking which shirts come from someone with Parkinson's disease?

Yes, we have very strong evidence that Milne tends to do better than just guessing because 11 correct out of 12 attempts is very unlikely to happen by chance alone

Suppose you are testing the hypotheses: H0: π = 0.60 and Ha: π > 0.60 and your sample proportion is 0.61. From this you determine that your standardized statistic is 0.53. What is the correct interpretation of this standardized statistic? --Your observed statistic of 0.53 is 0.61 standard deviations above the mean of the null distribution. --Your observed statistic of 0.61 is 0.53 standard deviations above the mean of the null distribution. --Assuming the null hypothesis is true, the probability you would get an observed statistic of 0.61 or more is 0.53. --Assuming the alternative hypothesis is true, the probability you would get an observed statistic of 0.61 or more is 0.53.

Your observed statistic of 0.61 is 0.53 standard deviations above the mean of the null distribution

Suppose a friend of yours says she is a 75% free-throw shooter in basketball. You don't think she is that good and want to test her to gather evidence that she makes less than 75% of her free throws in the long run. You have her shoot 40 free throws and she makes 26 (or 65%) of them. Set up the correct null and alternative hypotheses for this test. Which of the following is the appropriate sample statistic and proper symbol? ^P=0.75 pi = 0.75 ^p=o.65 pi=0.65 n=26

^p = 0.65

What does Ha represent?

alternative hypothesis

Statistic

correct object/# of trials

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 12 out of 20, or 60%, of his serves in. Jerry wants to know if his long-run proportion of getting his first serve in is greater than 0.50. What values would you enter into the One Proportion applet to run an analysis for Jerry? Assuming Jerry's long-run proportion of getting his new first serve is truly 0.50, 12 out of 20 is a(n) _______ value because it occurred frequently in the simulated data.

likely

A chimpanzee named Sarah was the subject in a study of whether chimpanzees can solve problems. Sarah was shown 30-second videos of a human actor struggling with one of several problems (for example, not able to reach bananas hanging from the ceiling). Then Sarah was shown two photographs, one that depicted a solution to the problem (like stepping onto a box) and one that did not match that scenario. Researchers watched Sarah select one of the photos, and they kept track of whether Sarah chose the correct photo depicting a solution to the problem. Sarah chose the correct photo in seven of eight scenarios that she was presented. The symbol π in this context stands for the _____ of times that Sarah chooses the correct photo

long-run proportion

What does pi represent?

long-run proportion (parameter)

What does Ho represent?

null hypothesis

Many types of studies have been done to see whether animals have a sense of number. In one study (Beran & Beran, 2004), researchers wanted to see whether chimpanzees could remember how many items were added to a container when they see them added one at a time. To do this they had two containers set up so the chimpanzees could not see inside them. They randomly added bananas to the containers one at a time and then had the chimpanzees choose the container they wanted. In one set of experiments, they added 3 bananas to one container and 4 to the other and had a chimpanzee named Mercury pick which container he wanted. In 20 trials, he picked the container containing 4 bananas 16 times (or 80%). Do these data convince you that Mercury is more likely to pick the container with more bananas in it in the long run? A null distribution, with 100 repititions, for testing this situation is shown. Null hypothesis: ALternative hypotehsis:

null: mercury will choose a random container alternative; mercury will choose the container with the most amount of bananas

How to find P

number of observed/ total

Many types of studies have been done to see whether animals have a sense of number. In one study (Beran & Beran, 2004), researchers wanted to see whether chimpanzees could remember how many items were added to a container when they see them added one at a time. To do this they had two containers set up so the chimpanzees could not see inside them. They randomly added bananas to the containers one at a time and then had the chimpanzees choose the container they wanted. In one set of experiments, they added 3 bananas to one container and 4 to the other and had a chimpanzee named Mercury pick which container he wanted. In 20 trials, he picked the container containing 4 bananas 16 times (or 80%). Do these data convince you that Mercury is more likely to pick the container with more bananas in it in the long run? A null distribution, with 100 repititions, for testing this situation is shown. Based on the null distribution shown (that has 100 repetitions), what is the p-value for your test?

p-value = 2/100 = 0.02

What does ^p represent?

sample proportion

What does n represent

sample size

The p-value of a test of significance is:

the probability, assuming the null hypothesis is true, that we would get a result at least as extreme as the one that was actually observed

Tennis players often spin a racquet to decide who serves first. The spun racquet can land with the manufacturer's label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down. Which of the following describes the relevant long-run proportion of interest?

the true proportion of times the racquet lands face up

Which of the following is the most important reason that a simulation analysis would repeat the coin-flipping process many times?

to see how much variability results in the distribution of sample proportions.

Jerry is a tennis player. He is working on a really tough first serve. While practicing his new tough serve, he gets 60 out of 100, or 60%, of his serves in. Jerry wants to know whether his long-run proportion of getting his first serve in is greater than 0.50. Use the One Proportion applet to test if Jerry's long-run proportion of getting his first serve in is greater than 0.50. State the values you would enter into the applet. If Jerry's long-run proportion of getting his first serve in is truly 0.50, 60 out of 100 is somewhat _______ because it occurred somewhat ________.

unlikely, infrequently

Suppose you are conducting a test of significance to try to determine whether your cat, Hope, will go to the correct object (out of two) when it is pointed to, just like Harley the dog did in the Exploration 1.1. You test Hope 100 times and finds she goes to the correct object in 70 of those 100 trials. In this scenario, what is the variable?

whether Hope goes to the correct object