Introduction to statistical investigations tintle pdf download

These are the two possible explanations to be evaluated. But does this seem like a reasonable explanation to you? How would you argue against someone who thought this was the case?

Is 15 out of 16 correct pushes convincing to you? Or do you think that Buzz could have just been guessing? How might you justify your answer? So how are we going to decide between these two possible explanations?

One approach is to choose a model for the random process repeated attempts to push the correct button and then see whether our model is consistent with the observed data. If it is, then we will conclude that we have a reasonable model and we will use that model to answer our questions. Statisticians often employ chance models to generate data from random processes to help them in- vestigate such processes. You did this with the Monty Hall exploration Section P.

In that exploration it was clear how the underlying chance process worked, even though the probabilities themselves were not obvious. We are trying to decide whether the process could be Buzz simply guessing or whether the process is something else, such as Buzz and Doris being able to communicate.

Because Buzz is choosing between two options, the simplest chance model to consider is a coin flip. We can flip a coin 1. The correspondence between the real study and the physical simulation is shown in Table 1. Imagine that we get heads on the first flip. What does this mean? This would correspond to Buzz pushing the correct button! But, why did he push the correct button? In this chance model, the only reason he pushed the correct button is be- cause he happened to guess correctly—remember the coin is simulating what happens when Buzz is just guessing which button to push.

What if we keep flipping the coin? Each time we flip the coin we are simulating another attempt where Buzz guesses which button to push. How many times do we flip the coin? After 16 tosses, we obtained the sequence of flips shown in Figure 1. Here we got 11 heads and 5 tails 11 out of 16, or 0. This gives us an idea of what could have happened in the study if Buzz had been randomly guess- ing which button to push each time.

Will we get this same result every time we flip a coin 16 times? When we did this, we got 7 heads and 9 tails, as shown in the sequence of coin flips 7 out of 16, or 0. So can we learn anything from these coin tosses when the results vary between the sets of 16 tosses?

But are some numbers of heads more likely than others? If we continue our repetitions of 16 tosses, we can start to see how the outcomes for the number of heads are distributed. Does the distribution of the number of heads that result in 16 flips have a predictable long-run pattern?

In particular, how much variability is there in our simulated statistics between repetitions sets of 16 flips just by random chance? In order to investigate these questions, we need to continue to flip our coin to get many, 1.

We did this, and Figure 1. Here, the process of flipping a coin 16 times was repeated times in Figure 1. For these graphs, each dot represents the number of heads in one set of 16 coin tosses.

We see that the resulting number of heads follows a clear pattern: 7, 8, and 9 heads happened quite a lot, 10 was pretty common also though less so than 8 , 6 happened some of the time, 1 happened once.

But we never got 15 heads in any set of 16 tosses! We might consider any outcome between about 5 and 11 heads to be typical, but getting fewer than 5 heads or more than 11 heads happened rarely enough we can consider it a bit unusual.

What does this have to do with the dolphin communication study? We said that we would flip a coin to simulate what could happen if Buzz was just guessing each time he pushed the button in 16 attempts. We saw that getting results like 15 heads out of 16 never happened in our 1, repetitions. This shows us that 15 is a very unusual outcome—far out in the tail of the distribution of the simulated statistics—if Buzz is just guessing.

In short, even though we expect some variability in the results for different sets of 16 tosses, the pattern shown in this distribution indicates that an outcome of 15 heads is outside the typical chance variability we would expect to see when Buzz is simply guessing.

In the actual study, Buzz really did push the correct button 15 times out of 16, an outcome that we just determined would rarely occur if Buzz was just guessing. So, our coin flip chance 1. The results mean our Definition evidence is strong enough to be considered statistically significant.

The steps we went through above have helped us evaluate how strong the evidence is that If our observed result Buzz is not guessing Step 4 of the statistical investigation method.

And if so, what does this say about other dolphins? After completing Steps 1—5 of the statistical investigation method, we need to revisit the big picture of the initial research question. First, we reflect on the limitations of the analysis and think about future studies. In short, we are now stepping back and thinking about the initial research question more than the specific research con- jecture being tested in the study.

For this study, we would reflect on Dr. The 3S strategy 1. We observed a sample statistic e. If it is unusual—we say the observed statistic is statistically significant—it provides strong evidence that the chance-alone explanation is wrong.

If it is typical, we consider the chance model plausible. You may have noticed that we only simulated results for one specific model. When we saw that the sample statistic observed in the study was not consistent with these simulated results, we rejected the chance-alone explanation.

Often, research analyses stop here. Instead of trying to simulate results from other models in particular we may not really have an initial idea what a more appropriate model might be , we are content to say there is something other than random chance at play here. This might lead the researchers to reformulate their conjec- tures and collect more data in order to investigate different models.

We will call the process of simulating could-have-been statistics under a specific chance model the 3S strategy. After forming our research conjecture and collecting the sample data, we will use the 3S strategy to weigh the evidence against the chance model.

This 3S strategy will serve as the foundation for addressing the question of statistical significance in Step 4 of the statistical investigation method. Statistic: Compute the statistic from the observed sample data. Repeatedly simu- late values of the statistic that could have happened when the chance model is true.

Strength of evidence: Consider whether the value of the observed statistic from the research study is unlikely to occur when the chance model is true. If we decide the observed statistic is unlikely to occur by chance alone, then we can conclude that the observed data provide strong evidence against the plausibility of the chance model.

If not, then we consider the chance model to be a plausible believable explanation for the observed data; in other words what we observed could plausibly have happened just by random chance. Statistic: Our observed statistic was 15, the number of times Buzz pushed the correct button in 16 attempts. Simulate: If Buzz was actually guessing, the parameter the probability he would push the correct button would equal 0. We used a coin flip to model what could have happened in 16 attempts when Buzz is just guessing.

We then repeat this process many more times, each time keeping track of the number of the 16 attempts that Buzz pushed the correct button. We end up with a distribution of could-have-been statistics representing typical values for the number of correct pushes when Buzz is just guessing. Strength of evidence: Because 15 successes in 16 attempts rarely happens by chance alone, we conclude that we have strong evidence that, in the long-run, Buzz is not just guessing.

Another Doris and Buzz study One goal of statistical significance is to rule out random chance as a plausible believable explanation for what we have observed. We still need to worry about how well the study was conducted. Are we sure there was no pattern to which headlight setting was displayed that he might have detected? But the chance of his being that lucky is so small that we conclude that other explanations are more plausible or credible.

One option that Dr. Bastian pursued was to redo the study except now he replaced the curtain with a wooden barrier between the two sides of the tank in order to ensure a more complete separation between the dolphins to see whether that would diminish the effective- ness of their communication.

The research question remains the same: Can dol- phins communicate in a deep abstract manner? The study design is similar with some ad- justments to the barrier between Doris and Buzz. The canvas curtain is replaced by a plywood board.

The research conjecture, observational units, and variable remain the same. In this case, Buzz pushed the correct button only 16 out of 28 times. The variable is the same whether or not Buzz pushed the correct button , but the number of observational units sample size has changed to 28 the number of attempts. Or is it believable that Buzz could have just been guessing? STEP 3: Explore the data.

So our observed statistic is 16 out of 28 correct attempts, 0. A simple bar graph of these results is shown in Figure 1. STEP 4: Draw inferences. Is it plausible believable that Buzz was simply guessing in 0. How do we measure how much evidence these results provide against the chance model? We will apply the 3S strategy to this new study.

Statistic: The new observed sample statistic is 16 out of 28, or about 0. Consider again our simulation of the chance model assuming Buzz is guessing. Simulation: This time we need to do repetitions of 28 coin flips, not just A distribution of the number of heads in 1, repetitions of 28 coin flips is shown in Figure 1.

This models 1, repetitions of 28 attempts with Buzz randomly pushing one of the buttons guessing each time. Lower tail Typical outcomes Upper tail. This models the number of correct pushes in 28 attempts when Buzz is guessing each time. Strength of evidence: Now we need to consider the new observed statistic 16 out of 28, or 0. We see from the graph that 16 out of 28 is a fairly typical outcome if Buzz is just ran- domly guessing.

What does this tell us? It tells us that the results of this study are something that could easily have happened if Buzz was just randomly guessing. So what can we conclude?

The graph in Figure 1. But be careful: The opposite result—an actual outcome near the center—is not strong evidence in support of the guessing hypothesis. Yes, the result is consistent with that hypothesis, but it is also consistent with many other hypotheses as well.

Bottom line: In this second study we conclude that there is not enough evidence that the 1. That model is still a plausible explanation for the statistic we observed in the study 16 out of Based on this set of attempts, we do not have con- vincing evidence against the possibility that Buzz is just guessing, but other explanations also remain plausible. For example, the results are consistent with very weak communication between the dolphins.

All we know from this analysis is that one plausible explanation for the observed data is that Buzz was guessing. In fact, Dr. Bastian soon discovered that in this set of attempts the equipment malfunc- tioned and the food dispenser for Doris did not operate and so Doris was not receiving her fish rewards during the study.

Bastian fixed the equipment and ran the study again. This time he found convincing evidence that Buzz was not guessing.

For a bit more discussion on processes and parameters, see FAQ 1. In that time, have they been able to develop an understanding of human gestures such as pointing or glancing?

How about similar nonhuman cues? Researchers Udell, Giglio, and Wynne tested a small number of dogs in order to answer these questions. In this exploration, we will first see whether dogs can understand human gestures as well as nonhuman gestures. To test this, the researchers positioned the dogs about 2.

On each side of the experimenter were two cups. The experimenter would perform some sort of gesture pointing, bowing, looking toward one of the cups or there would be some other nonhuman gesture a mechanical arm pointing, a doll pointing, or a stuffed animal looking toward one of the cups. The researchers would then see whether the dog would go to the cup that was indicated.

There were six dogs tested. We will look at one of the dogs in two of his sets of trials. This dog, a four-year-old mixed breed, was named Harley. Each trial involved one gesture and one pair of cups, with a total of 10 trials in a set. We will start out by looking at one set of trials where the experimenter bowed toward one of the cups to see whether Harley would go to that cup.

Harley was tested 10 times and 9 of those times he chose the correct cup. Identify the variable in the study. What are the possible outcomes of this variable? Is this variable quantitative or categorical? The set of observational 4. What is the number of observational units sample size in this study? Determine the observed statistic and produce a simple bar graph of the data have one bar collect data is called the for the proportion of times Harley picked the correct cup and another for the proportion sample.

The number of of times he picked the wrong cup. If the research conjecture is that Harley can understand what the experimenter means sample is the sample when they bow toward an object, is the statistic in the direction suggested by the research size.

A statistic is a conjecture? Do you think it is likely Harley would have gotten 9 out of 10 correct if he was just guess- ing randomly each time? There are two possibilities for why Harley chose the correct cup 9 out of 10 times:. That is, he got more than half correct just by random chance alone. The unknown long-run proportion i. What is the value of the parameter if Harley is picking a cup at random? Give a specific process.

What is the possible range of values greater than or less than some value for the para- meter if Harley is not just guessing and instead understands the experimenter? The chance model Statisticians often use chance models to generate data from random processes to help them investigate the process.

In particular, they can see whether the observed statistic is consistent with the values of the statistic simulated by the chance model. How many times do you have to flip the coin to represent chance.

What does heads represent? If Harley was guessing randomly each time, on average, how many out of the 10 times model, we say that the would you expect him to choose the correct cup? Simulate one repetition of Harley guessing randomly by flipping a coin 10 times why 10? Count the number of heads in your 10 flips. Combine your results with the rest of the class to create a dotplot of the distribution for the number of heads out of 10 flips of a coin.

Where does 9 heads fall in the distribution? Would you consider it an unusual outcome or a fairly typical outcome for the number of heads in 10 flips? Based on your answer to the previous question, do you think it is plausible believable that Harley was just guessing which cup to choose? Using an applet to simulate flipping a coin many times To really assess the typical values for the number of heads in 10 coin tosses number of correct picks by Harley assuming he is guessing at random , we need to simulate many more outcomes of the chance model.

Open the One Proportion applet from the textbook webpage. Notice that the probability of heads has been set to be 0. Set the number of tosses to 10 and press the Draw Samples button. What was the resulting number of heads? Notice that the number of heads in this set of 10 tosses is then displayed by a dot on the graph. Uncheck the Animate box and press the Draw Samples button 9 more times. This will demonstrate how the number of heads varies randomly across each set of 10 tosses.

Nine more dots have been added to your dotplot. Is a pattern starting to emerge? Now change the Number of repetitions from 1 to and press Draw Samples. The ap- plet will now show the results for the number of heads in 1, different sets of 10 coin tosses.

So each dot represents the number of times Harley chooses the correct cup out of 10 attempts assuming he is just guessing. Locate the result of getting 9 heads in the dotplot created by the applet. Would you con- sider this an unlikely result in the tail of the distribution of the number of heads? Do the results of this study appear to be statistically significant? Do the results of this study suggest that Harley just guessing is a plausible explanation for Harley picking the correct cup 9 out of 10 times?

Summarizing your understanding To make sure that you understand the coin-flipping chance model, fill in Table 1. The 3S strategy We will call the process of simulating could-have-been statistics under a specific chance model the 3S strategy.

This 3S strategy will serve as the foundation for addressing the question of statistical significance in Step 4 of the statisti- cal investigation method. Strength of evidence: Consider whether the value of the observed statistic from the research study is unlikely to occur if the chance model is true.

What is the statistic in this study? Fill in the blanks to describe the simulation. Strength of evidence. Fill in the blanks to summarize how we are assessing the strength of evidence for this study. Based on this analysis, are you convinced that Harley can understand human cues? Why or why not? Another study One important step in a statistical investigation is to consider other models and whether the results can be confirmed in other settings.

In a different study, the researchers used a mechanical arm roughly the size of a human arm to point at one of the two cups. The researchers tested this to see whether dogs understood nonhuman gestures. In 10 trials, Harley chose the correct cup 6 times. Using the dotplot you obtained when you simulated 1, sets of 10 coin flips assuming Harley was just guessing, locate the result of getting 6 heads.

Would you consider this an unlikely result in the tail of the distribution? Based on the results of 1, simulated sets of 10 coin flips each, would you conclude that Harley would be very unlikely to have picked the correct cup 6 times in 10 at- tempts if he was randomly guessing between the two cups each time? Do the results of this study suggest that Harley just guessing is a plausible explanation for Harley picking the correct cup 6 out of 10 times?

Does this study prove that Harley cannot understand the mechanical arm? Compare the analyses between the two studies. How does the unusualness of the observed statistic compare between the two studies? Does this make sense based on the value of the observed statistic in the two studies? Does this make sense based on how the two studies were designed? Hint: Why might the results differ for human and mechanical arms?

Why would this matter? A single study will not provide all of the information needed to fully understand a broad, complex research question.

Thinking back to the original research question, what addi- tional studies would you suggest conducting next? The number of ob- servational units is the sample size. A number computed to summarize the variable measured on a sample is called a statistic. For a chance process, a parameter is a long-run numerical property of that process, such as a probability long-run proportion.

A simulation analysis based on a chance model can assess the strength of evidence provided by sample data against a particular claim about the chance model. The logic of as- sessing statistical significance employs what we call the 3S strategy:.

Repeatedly simulate values of the statistic that could have occurred from that chance model. The chance model considered in this section involved tossing a fair coin. In the next section you will consider other chance models, but the reasoning process will remain the same. Not all situations call for a coin-flipping model.

Other events, as you will soon see, can be a bit more complicated than this. These things will both help us formalize the procedure of a test of significance and give us some guidelines to help us determine when we have strong enough evidence that our chance model is not correct.

We will also introduce some new symbols, for convenience, so try to keep the big picture in mind. Rock-Paper-Scissors 1. Official rules are available from the World Example 1. But is it really a fair game? Do some players exhibit patterns in their behavior that an opponent can exploit? An article published in College Mathematics Journal Eyler, Shalla, Doumaux, and McDevitt, found that players, particularly novices, tend to not prefer scissors.

You explain the rules of the game and play 12 rounds. Suppose your friend only shows scissors twice in those 12 plays. In this study, the individ- Definition ual plays of the game are the observational units, and the variable is whether or not the player chooses scissors. This categorical variable has just two outcomes scissors or not scissors A binary variable is a and so is sometimes called a binary variable.

Often game. The parameter of interest is the long-run proportion that any player picks scissors. Your friend only chose scissors one-sixth of the time. But perhaps that was just an unlucky occurrence in this study? Maybe your friend would play scissors one-third of the time if he played the game for a very long time, and, just by chance, you happened to observe less than one-third in the first 12 games?

In fact, we should state the hypotheses prior to conducting explanation that contra- the study, before we ever gather any data! Our goal is to use the sample data to estimate the dicts the null hypothesis. Throughout the data they collect. The distinction be- tween parameter and statistic is so important that we always use different symbols to refer to them.

In fact, one way to distinguish between the parameter and the statistic is verb tense! The statistic is the proportion of times that your friend did past tense, observed show scissors.

The parameter is the long-run proportion he would throw scissors future tense, unobserved if he played the game forever. We can also use symbols for the hypotheses. The null hypothesis is often written as H0 1. What differs between the two statements is the inequality symbol.

The null hypothesis will always contain an equals sign and the alternative hypothesis will contain a strictly greater than sign as it would have in the Doris and Buzz example from the previous question , a strictly less than sign as it does in this example , or a not equal to sign like we will see in Section 1. Which inequality symbol to use in the alternative hypothesis is determined by the research conjecture.

Statistic: Your friend showed scissors one-sixth of the time in the first 12 plays. Simulation: We will again focus on the chance-alone explanation, to see whether we have strong evidence that your friend chooses scissors less than one-third of the time in the long run.

If not, can you suggest a different random device we could use? Each chapter follows a coherent six-step statistical exploration and investigation method ask a research question, design a study, explore the data, draw inferences, formulate conclusions, and look back and ahead enabling students to assess a variety of concepts in a single assignment.

Challenging questions based on research articles strengthen critical reading skills, fully worked examples demonstrate essential concepts and methods, and engaging visualizations illustrate key themes of explained variation. The end-of-chapter investigations expose students to various applications of statistics in the real world using real data from popular culture and published research studies in variety of disciplines.

Accompanying examples throughout the text, user-friendly applets enable students to conduct the simulations and analyses covered in the book. Sold out. Students are actively engaged in the material through numerous activities in which they flip coins, draw cards, collect data, do computer simulations and run experiments. Computer simulations are done with freely available applets.

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