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The Test: Hypothesis Rising

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The Test: Hypothesis Rising is a video game developed and published by Randumb Studios . It was released on Steam for Microsoft Windows on April 27, 2020. It is the second game that is part of the The Test series .

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The second chapter in The Test trilogy. Answer a series of in-depth questions to reveal hidden truths about yourself and unlock secrets to discover how your life can change drastically with just a few, slight modifications. ...But are you sure you want the answers you seek?

The Test: Hypothesis Rising

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The Test: Hypothesis Rising is the second chapter in a three-part trilogy. The Test series is a social experiment that can be classified as an atmospheric, psychological-simulation with horror elements. Dive into your own psyche and get answers to questions you've kept hidden deep, down inside. Unlock your mind in ways never done before and experience new found knowledge simply by taking, " The Test: Hypothesis Rising ."

In the second chapter of The Test series, the questions get bolder, the stakes get higher, and the results get more specific as we uncover more and more about what makes you... Well... You.

You will answer a series of questions that will reveal things from deep within. These questions may make you uncomfortable, but with great knowledge comes great sacrifice. Can you make it all the way through The Test: Hypothesis Rising, and if you do, are you sure you want the answers you seek?## Features:

• Short and sweet
• Be a part of history with Player - Developer interaction
• The only series of its kind
• Very atmospheric and psychological
• Sit back, relax, and answer in-depth questions to reveal hidden truths
• Super affordable for a unique experience**

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• Game Engine: RPG Maker
• The Test series

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Average score: 2.0 out of 5 (based on 1 ratings with 0 reviews)

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Game added by BOIADEIRO ERRANTE .

Game added February 28, 2021. Last modified September 27, 2023.

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The Test: Hypothesis Rising

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Summary The second chapter in The Test trilogy. Answer a series of in-depth questions to reveal hidden truths about yourself and unlock secrets to discover how your life can change drastically with just a few, slight modifications. ...But are you sure you want the answers you seek?

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Hypothesis Testing Game

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This app quizzes your knowledge of hypothesis testing concepts using a tic-tac-toe game format.

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The Test (series)

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The Test series is a series of simulation adventure games developed and published by Randumb Studios. It is available for Microsoft Windows. The main characters are Choice, Fate and Destiny for now.

Games [ | ]

• The Test: Hypothesis Rising
• The Test: Final Revelation

The Test: Hypothesis Rising

• Short and sweet
• Be a part of history with Player - Developer interaction
• The only series of its kind
• Very atmospheric and psychological
• Sit back, relax, and answer in-depth questions to reveal hidden truths
• Super affordable for a unique experience

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 The Test: Hypothesis Rising

The second chapter in The Test trilogy. Answer a series of in-depth questions to reveal hidden truths about yourself and unlock secrets to discover how your life can change drastically with just a few, slight modifications. ...But are you sure you want the answers you seek?

Hey there, fellow gamers! Brace yourselves, because GameGal is about to blow your minds with a killer pitch for The Test: Hypothesis Rising! Picture this: you, yes YOU, embarking on a mind-bending adventure, where your very existence is put under the microscope. The Test trilogy continues with the second chapter, and boy, oh boy, are you in for a wild ride! So, the deal is simple, my pixelated pals. You'll be faced with a barrage of in-depth questions that will peel back the layers of your being, unveiling hidden truths and unlocking secrets you never knew existed. It's like a psychological deep dive into the very essence of who you are. Are you ready to unravel the mysteries of your own mind? But hold up, my eager players! Here's the twist. Are you sure you want the answers you seek? The Test: Hypothesis Rising isn't your average game; it's an enigma wrapped in a riddle, with a side of existential crisis. Yeah, you heard me right. You might think you're prepared for the truth bombs this game will drop on you, but can you handle it? Can you handle the truth? *Cue intense music* Oh, and did I mention the life-altering power of slight modifications? Your choices, big or small, can change the course of your virtual life forever. It's like playing God, but with a wicked gaming twist. Think of it as a Choose Your Own Adventure book, but on steroids, with a sprinkle of GameGal magic. Now, let's talk aesthetics, my fashion-forward amigos. This game takes you on a visual journey like no other. From the neon-lit cityscape to the mind-bending dreamscapes, The Test: Hypothesis Rising will blow your mind with its stunning graphics and vibrant art style. It's like stepping into a funky, futuristic world where every pixel is a work of freaking art. So, there you have it, my fierce gamers. The Test: Hypothesis Rising is an epic adventure that will challenge your mind, push your limits, and leave you questioning everything you thought you knew about yourself. Get ready to unlock secrets, make life-altering choices, and discover the true power of your own existence. Are you brave enough to face The Test? The world awaits, my friends. Game on!

~ GameGal, #AI #review #inaccurate #fun

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LESSWRONG LW

A hypothesis testing video game.

The Blob Family is a simple game made by Leon Arnott. At heart, it's a game about testing hypotheses and getting the right answer with the least amount of evidence you can.

The mechanics work like so: Balls bounce around the screen randomly and you control a character who needs to avoid them. You can aim the mouse anywhere and activate a sonar. On the right side are rules for how various balls will react to this, and your goal is to figure out which ball is which. As you use the sonar more, the balls speed up, so it becomes more difficult to stay alive, thus giving an incentive to test your hypothesis in as few clicks as possible.

It very nicely illustrates the principle that, to test a hypothesis, you must design tests to falisfy your intuitions rather than to confirm them. For example, in one level, when you use the sonar:

• 1 ball heads toward the center
• 1 ball heads away from the center
• 1 ball heads away from the mouse
• 1 ball heads away from you

I found myself mistakenly clicking in the center of the screen to test hypothesis 1, but this is insufficient. To design the proper tests, you need to keep the mouse out of the center, keep it away from you, and depending on the position of the balls keep it off a straight line from you.

It could also demonstrate the ability of a fast brain to test hypotheses quickly. For many levels, if you could slow time down and set up a very good test, you could solve the problem with a single click. But we humans aren't usually so attentive.

Just thought the LW crowd might enjoy it.

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Recently I've had another of my occasional flings with NetHack (warning: addictive). It strikes me as a great example of a computer game that teaches about probability, inference and hypothesis testing.

I'm assuming this only applies if you aren't using spoilers for NetHack?

I'm using all the spoilers I can find, and still find it a challenging game. Feel free to mock me. :)

A "spoiled" game of NetHack means you have precise numerical values of the upsides and downsides of various actions, e.g. rubbing a magic lamp. Or reading a scroll that you know from a shopkeeper's offer can be one of N scrolls, some of which have beneficial effects and others harmful. That definitely requires probabilistic decision-making - indulge in wishful thinking and you'll die often; play too cautiously (ignoring positive EVs of some actions with nasty downsides) and, well, you'll die often.

I suppose playing it "unspoiled" is even better, as you'd have to infer the frequencies from observation as opposed to having them delivered on a silver platter, as it were.

(ETA May 10th: finally Ascended as a Knight.)

Oh, no, I have no problems with people spoiling themselves for Nethack. That's pretty much the only way to actually win. But if your aim is to improve rationality, rather than to do as well as possible within the game, it might be better to play it unspoiled. After all, Morendil mentioned "hypothesis testing" as something that was taught by Nethack: The spoilt version doesn't really test that.

The spoilt version doesn't really test that.

It teaches the virtue of scholarship.

What I've found that the spoilt version of Nethack tests, more than anything else, is patience. Nethack spoilt isn't about scholarship, really. You don't study. You have a situation, and you look up things that are relevant to that situation. There is a small bit of study at the beginning, generally when you look up stuff like how to begin, what a newbie-friendly class/race is, and how to not die on the second floor.

But really, it's patience. I once did an experiment where players who were relatively new to Nethack were encouraged to spoil themselves as early and often as possible, and request advice frequently from better players. Really, to do anything short of having someone else play the game for you was not only allowed, but actively encouraged. Since I usually put a limiter on how willing I am to spoil myself on roguelikes, I thought this might be fun. (Namely, I'm unwilling to ask for any advice in tactical situations, only strategic ones: Which area should I go to next, instead of "How do I kill this ogre?")

Conventional wisdom for Nethack states that upon reaching the halfway point of the game, you should win from there if you play correctly. I got about three-quarters of the way there, on my third run, having never gotten past the second floor on my runs prior to those three. I died to a misclick, not to lack of knowledge or poor tactics. So, patience is the true virtue of Nethack: It's surprisingly easy to win as long as you spoil yourself, get advice, and don't screw up.

Sadly, the experiment only had the one participant actually try it, namely me, so the evidence shall remain anecdotal.

You can definitely only notice the rationality lessons if you already know them explicitly. I wonder how great the effect is when you're playing the game but don't know about the lessons. My guess: not very great. People can figure it out by instinct. It makes them use the hypothesis-testing skills they already have, not improve them. Most of the difficulty (for me at least) is not in the hypothesis testing, but in multitasking enough to find out what multiple balls do at once, then tracking them all as they bounce around the screen until you can tag them all.

Also, it's a short game, one you're unlikely to spend enough time doing for it to mold you. If you really want to get good at hypothesis testing, I recommend taking up programming. Debugging often stretches my hypothesis testing skills like nothing else I can think of.

I'm not sure about it's rationality testing or improving abilities, but I find it very fun :)

I doubt it will much improve anyone's rationality. It does nicely illustrate a few issues on how science is done, and could be a fun way of explaining for the layman.

Harriet (and xerxes acting like harriet) is bugged to always move away from me (at least when I run the game). I have to beat levels containing her by guessing :P

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Buy The Test

Buy the test trilogy bundle ().

Includes 6 items: The Test , The Test: Hypothesis Rising , The Test: Final Revelation , The Testing Chamber , The Test: Secrets of the Soul - Supporter Pack , The Test: Secrets of the Soul Supporter Pack 2

Buy Randumb Studios Deluxe Bundle BUNDLE (?)

Includes 26 items: The Summoning , The Twins , Naked News , The Test , The Horrorscope , The Test: Hypothesis Rising , The Confession , The Horrorscope: Fatal Awakening , Survive or Thrive , The Test: Final Revelation , Survivor Dieland , The Journey - Episode 1: Whatever This Is , The Advisor - Episode 1: Royal Pain , WTF is wrong with you? , The Questionably Quirky Quiz Show - Episode 1 , Gobby McGobblenutz Presents: The Art of the Dad Joke: Chapter 1 , Medieval Fantasy Survival Simulator , The Testing Chamber , WTF Do You Know? , Medieval Fantasy Survival Simulator 2: Gladiator Edition , The Journey - Episode 2: Words of Wisdom , Zombie Apocalypse Survival Simulator , Fateweaver: The Alchemist's Quandary , The Test: Secrets of the Soul - Supporter Pack , Fateweaver: Smash or Pass , The Test: Secrets of the Soul Supporter Pack 2

Buy The Test Trilogy Fateweaver Bundle BUNDLE (?)

Includes 5 items: The Test , The Test: Hypothesis Rising , The Test: Final Revelation , Fateweaver: The Alchemist's Quandary , Fateweaver: Smash or Pass

“That was fascinating! That was REALLY fascinating! That was crazy... I did not expect that going into it!” Markiplier “Rated 100th best Steam game of all time, with 98% positive reviews from 14,439 gamers!” https://club.steam250.com/app/1241510

About This Game

• Short and sweet
• The only game of its kind
• Very psychological
• Sit back, relax, and answer Yes or No questions to reveal hidden truths.
• Super affordable for a unique experience

System Requirements

• OS *: Windows 98/XP/Vista/7/8/10/11
• Processor: Intel Pentium III 800 Mhz
• Graphics: 1024x768 or better video resolution in High Color mode
• Storage: 700 MB available space
• Sound Card: DirectSound-compatible sound card

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Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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11.2: Correlation Hypothesis Test

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The correlation coefficient, \(r\), tells us about the strength and direction of the linear relationship between \(x\) and \(y\). However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient \(r\) and the sample size \(n\), together. We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute \(r\), the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, \(r\), is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is \(\rho\), the Greek letter "rho."
• \(\rho =\) population correlation coefficient (unknown)
• \(r =\) sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient \(\rho\) is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient \(r\) and the sample size \(n\).

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is "significant."

• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is significantly different from zero.
• What the conclusion means: There is a significant linear relationship between \(x\) and \(y\). We can use the regression line to model the linear relationship between \(x\) and \(y\) in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is "not significant".

• Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is not significantly different from zero."
• What the conclusion means: There is not a significant linear relationship between \(x\) and \(y\). Therefore, we CANNOT use the regression line to model a linear relationship between \(x\) and \(y\) in the population.
• If \(r\) is significant and the scatter plot shows a linear trend, the line can be used to predict the value of \(y\) for values of \(x\) that are within the domain of observed \(x\) values.
• If \(r\) is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If \(r\) is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed \(x\) values in the data.

PERFORMING THE HYPOTHESIS TEST

• Null Hypothesis: \(H_{0}: \rho = 0\)
• Alternate Hypothesis: \(H_{a}: \rho \neq 0\)

WHAT THE HYPOTHESES MEAN IN WORDS:

• Null Hypothesis \(H_{0}\) : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between \(x\) and \(y\) in the population.
• Alternate Hypothesis \(H_{a}\) : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between \(x\) and \(y\) in the population.

DRAWING A CONCLUSION:There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the \(p\text{-value}\)
• Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, \(\alpha = 0.05\)

Using the \(p\text{-value}\) method, you could choose any appropriate significance level you want; you are not limited to using \(\alpha = 0.05\). But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, \(\alpha = 0.05\). (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

METHOD 1: Using a \(p\text{-value}\) to make a decision

Using the ti83, 83+, 84, 84+ calculator.

To calculate the \(p\text{-value}\) using LinRegTTEST:

On the LinRegTTEST input screen, on the line prompt for \(\beta\) or \(\rho\), highlight "\(\neq 0\)"

The output screen shows the \(p\text{-value}\) on the line that reads "\(p =\)".

(Most computer statistical software can calculate the \(p\text{-value}\).)

If the \(p\text{-value}\) is less than the significance level ( \(\alpha = 0.05\) ):

• Decision: Reject the null hypothesis.
• Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is significantly different from zero."

If the \(p\text{-value}\) is NOT less than the significance level ( \(\alpha = 0.05\) )

• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is NOT significantly different from zero."

Calculation Notes:

• You will use technology to calculate the \(p\text{-value}\). The following describes the calculations to compute the test statistics and the \(p\text{-value}\):
• The \(p\text{-value}\) is calculated using a \(t\)-distribution with \(n - 2\) degrees of freedom.
• The formula for the test statistic is \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}\). The value of the test statistic, \(t\), is shown in the computer or calculator output along with the \(p\text{-value}\). The test statistic \(t\) has the same sign as the correlation coefficient \(r\).
• The \(p\text{-value}\) is the combined area in both tails.

An alternative way to calculate the \(p\text{-value}\) ( \(p\) ) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: \(p\text{-value}\) method

• Consider the third exam/final exam example.
• The line of best fit is: \(\hat{y} = -173.51 + 4.83x\) with \(r = 0.6631\) and there are \(n = 11\) data points.
• Can the regression line be used for prediction? Given a third exam score ( \(x\) value), can we use the line to predict the final exam score (predicted \(y\) value)?
• \(H_{0}: \rho = 0\)
• \(H_{a}: \rho \neq 0\)
• \(\alpha = 0.05\)
• The \(p\text{-value}\) is 0.026 (from LinRegTTest on your calculator or from computer software).
• The \(p\text{-value}\), 0.026, is less than the significance level of \(\alpha = 0.05\).
• Decision: Reject the Null Hypothesis \(H_{0}\)
• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score (\(x\)) and the final exam score (\(y\)) because the correlation coefficient is significantly different from zero.

Because \(r\) is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

METHOD 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of \(r\) is significant or not . Compare \(r\) to the appropriate critical value in the table. If \(r\) is not between the positive and negative critical values, then the correlation coefficient is significant. If \(r\) is significant, then you may want to use the line for prediction.

Example \(\PageIndex{1}\)

Suppose you computed \(r = 0.801\) using \(n = 10\) data points. \(df = n - 2 = 10 - 2 = 8\). The critical values associated with \(df = 8\) are \(-0.632\) and \(+0.632\). If \(r <\) negative critical value or \(r >\) positive critical value, then \(r\) is significant. Since \(r = 0.801\) and \(0.801 > 0.632\), \(r\) is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

Exercise \(\PageIndex{1}\)

For a given line of best fit, you computed that \(r = 0.6501\) using \(n = 12\) data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because \(r >\) the positive critical value.

Example \(\PageIndex{2}\)

Suppose you computed \(r = –0.624\) with 14 data points. \(df = 14 – 2 = 12\). The critical values are \(-0.532\) and \(0.532\). Since \(-0.624 < -0.532\), \(r\) is significant and the line can be used for prediction

Exercise \(\PageIndex{2}\)

For a given line of best fit, you compute that \(r = 0.5204\) using \(n = 9\) data points, and the critical value is \(0.666\). Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because \(r <\) the positive critical value.

Example \(\PageIndex{3}\)

Suppose you computed \(r = 0.776\) and \(n = 6\). \(df = 6 - 2 = 4\). The critical values are \(-0.811\) and \(0.811\). Since \(-0.811 < 0.776 < 0.811\), \(r\) is not significant, and the line should not be used for prediction.

Exercise \(\PageIndex{3}\)

For a given line of best fit, you compute that \(r = -0.7204\) using \(n = 8\) data points, and the critical value is \(= 0.707\). Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because \(r <\) the negative critical value.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example. The line of best fit is: \(\hat{y} = -173.51 + 4.83x\) with \(r = 0.6631\) and there are \(n = 11\) data points. Can the regression line be used for prediction? Given a third-exam score ( \(x\) value), can we use the line to predict the final exam score (predicted \(y\) value)?

• Use the "95% Critical Value" table for \(r\) with \(df = n - 2 = 11 - 2 = 9\).
• The critical values are \(-0.602\) and \(+0.602\)
• Since \(0.6631 > 0.602\), \(r\) is significant.
• Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score (\(x\)) and the final exam score (\(y\)) because the correlation coefficient is significantly different from zero.

Example \(\PageIndex{4}\)

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if \(r\) is significant and the line of best fit associated with each r can be used to predict a \(y\) value. If it helps, draw a number line.

• \(r = –0.567\) and the sample size, \(n\), is \(19\). The \(df = n - 2 = 17\). The critical value is \(-0.456\). \(-0.567 < -0.456\) so \(r\) is significant.
• \(r = 0.708\) and the sample size, \(n\), is \(9\). The \(df = n - 2 = 7\). The critical value is \(0.666\). \(0.708 > 0.666\) so \(r\) is significant.
• \(r = 0.134\) and the sample size, \(n\), is \(14\). The \(df = 14 - 2 = 12\). The critical value is \(0.532\). \(0.134\) is between \(-0.532\) and \(0.532\) so \(r\) is not significant.
• \(r = 0\) and the sample size, \(n\), is five. No matter what the \(dfs\) are, \(r = 0\) is between the two critical values so \(r\) is not significant.

Exercise \(\PageIndex{4}\)

For a given line of best fit, you compute that \(r = 0\) using \(n = 100\) data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between \(x\) and \(y\) in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between \(x\) and \(y\) in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatter plot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

The assumptions underlying the test of significance are:

• There is a linear relationship in the population that models the average value of \(y\) for varying values of \(x\). In other words, the expected value of \(y\) for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
• The \(y\) values for any particular \(x\) value are normally distributed about the line. This implies that there are more \(y\) values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of \(y\) values lie on the line.
• The standard deviations of the population \(y\) values about the line are equal for each value of \(x\). In other words, each of these normal distributions of \(y\) values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

Linear regression is a procedure for fitting a straight line of the form \(\hat{y} = a + bx\) to data. The conditions for regression are:

• Linear In the population, there is a linear relationship that models the average value of \(y\) for different values of \(x\).
• Independent The residuals are assumed to be independent.
• Normal The \(y\) values are distributed normally for any value of \(x\).
• Equal variance The standard deviation of the \(y\) values is equal for each \(x\) value.
• Random The data are produced from a well-designed random sample or randomized experiment.

The slope \(b\) and intercept \(a\) of the least-squares line estimate the slope \(\beta\) and intercept \(\alpha\) of the population (true) regression line. To estimate the population standard deviation of \(y\), \(\sigma\), use the standard deviation of the residuals, \(s\). \(s = \sqrt{\frac{SEE}{n-2}}\). The variable \(\rho\) (rho) is the population correlation coefficient. To test the null hypothesis \(H_{0}: \rho =\) hypothesized value , use a linear regression t-test. The most common null hypothesis is \(H_{0}: \rho = 0\) which indicates there is no linear relationship between \(x\) and \(y\) in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

Formula Review

Least Squares Line or Line of Best Fit:

\[\hat{y} = a + bx\]

\[a = y\text{-intercept}\]

\[b = \text{slope}\]

Standard deviation of the residuals:

\[s = \sqrt{\frac{SSE}{n-2}}\]

\[SSE = \text{sum of squared errors}\]

\[n = \text{the number of data points}\]

Carl Willis’ message to starters: Throw strikes, don’t press — Guardians takeaways

• Updated: Apr. 28, 2024, 5:44 p.m. |
• Published: Apr. 28, 2024, 3:11 p.m.

Pitching coach Carl Willis (right) talks to right-hander Triston McKenzie. Joshua Gunter, cleveland.com

• Paul Hoynes, cleveland.com

ATLANTA — If the Guardians ’ rotation is going to pitch deeper into games, Tanner Bibee paved the way Saturday night against the Braves.

Bibee turned in the longest start by a Cleveland pitcher this season when he went seven scoreless innings in a 4-2, 11-inning win over the Braves . He struck out nine and allowed two hits on 91 pitches.

More Guardians coverage

• Cleveland Guardians, Atlanta Braves lineups for April 28, 2024: Game 28
• Guardians’ Josh Naylor craves the big moment in a humble way
• Guardians lose one late lead, but forge another in 4-2 win over Braves in 11 innings

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Carlos Santana, Twins hammer Angels 16-5 for sixth straight win

Santana homered for the third time in as many games as the Twins got back to .500.

• Author: Nolan O'Hara

In this story:

This is what the Twins were hoping for when they signed Carlos Santana. 

Santana hit his third home run in as many games, this time a three-run shot in the fourth inning, to power the Twins past the Los Angeles Angels 14-5 Saturday night in Anaheim, Calif. 

The Twins are back to .500 at 13-13 after a six-game winning streak. 

Santana hit his first homer of the year in Thursday’s victory in the series finale against the Chicago White Sox. He hit another in Friday night’s series-opening victory over the Angels. Then on Saturday, he took four-seamer from Angels reliever Jose Cisnero 408 feet into the right-field bleachers for a three-run homer that gave the Twins a 10-4 lead in the fourth inning. 

That scored Trevor Larnach and Will Castro, who were walked and hit a single, respectively. 

Santana was 2 for 4 overall Saturday night with four RBIs and three runs scored.

The Twins chased Angels starter Jose Soriano early. In the first inning, Alex Kirilloff drew a leadoff walk before Edouard Julien advanced him to third base with a single to right field.

Ryan Jeffers was then hit by a pitch to load the bases. 

After Trevor Larnach struck out, Max Kepler drew another walk to force in the opening run of the game. While Soriano got out of the inning with minimal damage after getting Willi Castro to ground into a 6-3 double play, his day wouldn’t last too much longer. 

Soriano hit Santana with a pitch to open the second inning before walking Kyle Farmer. Austin Martin grounded out to short but advanced Santana and Farmer to second and third, and Kirilloff scored Santana with a sacrifice fly to center field to make it 2-0. 

Then Julien scored Farmer with a single to left field, and Jeffers followed up with another single to put runners at first and second. That put an end to Soriano’s day as Angels manager Ron Washington turned to Jose Suarez, who gave up a single to Larnach that scored Julien for a 4-0 Twins lead, although the inning would end when Jeffers was thrown out at third base. 

Jo Adell hit a two-run homer in the second inning off Twins starter Chris Paddack to put the Angels (10-17) back within two runs, but the Twins just kept stringing hits together and adding more runs.

Paddack went five innings Saturday, allowing eight hits and four runs while fanning a pair.

Kepler and Castro hit back-to-back singles to open the third inning before Santana and Farmer hit back-to-back doubles, Santana’s scoring Kepler and Farmer’s scoring both Castro and Santana. That gave the Twins a 7-2 lead before Suarez retired the next three Twins batters in order. 

The Angels got two back in the bottom of the frame, getting RBI singles from Aaron Hicks and Adell. But that set the stage for Santana’s three-run blast in the fourth inning to make it 10-4.

Kepler later hit a solo homer that went 407 feet into center field in the sixth inning to make it 11-4, and the Twins plated three more in the seventh inning off an RBI double from Kirilloff and a two-RBI double from Julien. It was just an absolute beating from the Twins.

Kody Funderburk pitched a 1-2-3 sixth inning, but he did give up a run in the seventh on a sacrifice fly from Brandon Drury after walking Mike Trout and giving up a single to Taylor Ward. Funderburk allowed just the one hit and one run while fanning one across two innings.

Jeffers hit a two-run homer in the ninth inning off Hicks, the designated hitter and outfielder who was pitching in the ninth inning to save a bullpen arm for the Angels, to add further insult to injury.

Jay Jackson pitched scoreless eighth and ninth innings to close it out for the Twins.

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2024 NFL Draft: Broncos make history by reuniting first-round pick Bo Nix with his favorite Oregon playmaker

A singular nfl team took a qb and wr from the same school for just the third time in league history.

The Denver Broncos took Oregon quarterback Bo Nix No. 12 overall on the first night of the 2024 NFL Draft . With one of the first picks on Day 3, the organization took one of his favorite pass catchers: wide receiver Troy Franklin. 

It is only the third time in the common draft era that a singular team selected a quarterback and wide receiver from the same school. The other two occurrences were in 1979 when the Chiefs took Clemson quarterback Steve Fuller and wide receiver Stan Rome and in 2019 when the Commanders drafted Ohio State quarterback Dwayne Haskins and wide receiver Terry McLaurin . 

Franklin departed Oregon as the program's all-time leader in receiving touchdowns (25) and tied for the most games with 100+ yard receiving. In his 2023, his 1,383 receiving yards and 14 touchdowns were single-season records. 

Head coach Sean Payton has put his fingerprints on the Denver organization since arriving last winter. Quarterback Russell Wilson and wide receiver Jerry Jeudy have been released and traded, respectively. The Broncos took Marvin Mims Jr. in the second round a year ago and still employ Courtland Sutton and K.J. Hamler . 

Sandwiched between Nix and Franklin was Utah pass rusher Jonah Elliss . 

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The Test: Hypothesis Rising

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