Understanding Hypothesis Testing in Military Statistics

Hypothesis testing determines whether to accept or reject established theories using statistical methods.

### Understanding Hypothesis Testing: Concepts and Applications
* 📊 In Week 11 of Military Statistics, we delve into hypothesis testing, focusing on the **null and alternative hypotheses**, **significance levels**, and various testing methods such as **right-tailed and left-tailed tests**.
* 🔍 A hypothesis is a testable statement about the relationship between variables, with the **null hypothesis** representing the existing theory that researchers aim to **challenge**, while the **alternative hypothesis** suggests that the null is **incorrect**.
* 📈 The testing process involves calculating a **test statistic** and determining whether it falls within a **confidence interval**, which helps in deciding whether to **accept or reject the null hypothesis** based on the **significance level**, typically set below **0.1**.
* ⚖ The significance level, often denoted as alpha (10%, 5%, or 1%), indicates the **maximum probability of making an error when rejecting the null hypothesis**, with common values being **less than 0.1 for acceptable error rates**.
* 📏 In hypothesis testing, researchers evaluate whether sample statistics fall within a **predetermined range** to support the null hypothesis or indicate the validity of the research hypothesis, leading to a **critical decision-making process**.
* 📉 The p-value represents the **probability of obtaining a test statistic that is more extreme than the observed value**, guiding researchers in determining the **accuracy of their hypotheses**.
* 🧮 Through examples, such as measuring star candies in gun bread, we demonstrate how to apply hypothesis testing to real-world scenarios, emphasizing the **importance of understanding statistical values and their implications**.
* 📚 As we conclude, remember that **mastering hypothesis testing concepts** will greatly aid your analytical skills, and I encourage you to **review the material thoroughly** for a deeper understanding.

Hello, I am Jeong-yi from Military Statistics. This time, we are in Week 11, and the topic is the concept of hypothesis testing. Without further ado, let's start. 

The first topic of the lecture is the types of hypotheses and how to determine the significance level. We will study the process of hypothesis testing. Here are the details of today’s lecture: we will examine the null hypothesis and alternative hypothesis, as well as special tests, including right-tailed tests and left-tailed tests. We will also study how to calculate the critical value and decide whether to accept or reject the hypothesis.

The concept of a hypothesis refers to a testable statement about the relationship between variables or phenomena, framed in a way that can be scientifically investigated. If I say this, it might sound confusing, but hypotheses are something you often encounter, right? I hope it will be simple to understand. We have a straightforward framework, and as we study, I will explain the hypotheses one by one.

In statistics, research hypotheses can be broadly categorized into two types: the null hypothesis and the alternative hypothesis. I will first explain the null hypothesis to you. The null hypothesis refers to the existing theory or statement that the researcher aims to challenge. Generally, it is expressed in terms of a set of parameters or conditions. The existing theories, which we refer to as the null hypothesis, are the established beliefs or assumptions we have held until now. 

The research hypothesis, on the other hand, posits that the null hypothesis is not valid. Essentially, the researcher is challenging the established theory, suggesting that it may not be correct. Since this is a challenge, it necessitates testing. Therefore, we need to conduct tests to determine which hypothesis is accurate and verify the facts.

It’s interesting, isn’t it? The hypothesis is intriguing. The testing is conducted under the premise that the null hypothesis is always correct until the scale of the hypothesis is rejected. The test statistic is calculated, and the testing process is carried out. This graph is something we've seen a lot in textbooks, right? We’ve seen this up to this point. Now, what do we have here? We have confidence interval levels. If we set a standard sample and improve it, and the numbers fall within this range, then the existing theory is correct. The champion is omnipotent. However, if the numbers fall outside, then the champion is wrong, and the challenger is correct. That's how it works. If the null hypothesis is true, then if it falls within this range, the null hypothesis is accepted, while anything outside the confidence level is considered an error.

If it falls within the range, the research hypothesis is adopted. When we measured the blood pressure of the students in our class and department, we found that they all showed high levels. I’m not sure about the details, but is a reading between 130 and 152 considered normal? We thought it was normal. However, after graduation, when we saw readings below 100, it didn’t seem right. It felt like it wasn't normal. When we checked again, we found that there could be variations in the readings. This is directly related to the concept of the hypothesis. Under the premise that the hypothesis is correct, if we obtain somewhat extreme statistical values, we may recall the scale hypothesis and consider adopting the research hypothesis. The maximum probability of obtaining extreme statistical values is referred to as the significance level, which we usually set to be less than 0.1. However, what have we been calling this so far? We should refer to it as the margin of error.

According to the allowable range, we can determine this well. The probability of this happening is 90%, 95%, or 99%. The values outside of this range are what we refer to as alpha. So, what is alpha? It is 10%, 5%, or 1%. Specifically, 10% corresponds to 0.1, 5% to 0.05, and 1% to 0.01. These values are small. Therefore, the general level of acceptable error for alpha is mostly below 0.1, meaning it is less than 90%. 

In the case of a research hypothesis being tested, the probability that the challenger can defeat the champion hypothesis, under the premise that the scale is correct, is only at the u level. This might be difficult to understand conceptually, but as we study through examples, it will become easier to grasp. We will evaluate and judge whether a new hypothesis can be considered correct using statistical methods.

It is said that this is a process. We talk about a group being a population. So, how do we determine if it is correct or not? We need to select a sample. If the sample falls within the acceptable range, it supports the existing hypothesis; if it goes outside, it indicates that the research hypothesis may be incorrect. We are verifying whether the hypothesis is right or wrong through this process. We studied the allowable error range earlier, right? Under the assumption that the hypothesis is correct, if we get a sample statistic that is somewhat extreme, we will judge that the hypothesis is incorrect and reject it. The maximum error rate will be set at a certain level. We will also discuss this further later.

The fishing vessel is simply studying terminology. In mathematics or interpretation, the terminology is less than half. Regarding threshold validation, there are types that are two-sided and ones that are shortened. Let me explain how this works. Considering the angle of incidence, it determines whether to accept the hypothesis based on the boundary. In the March issue, we study the threshold once. When I explain this, I mention the term critical angle. There is water at the critical angle, and if we attach a medium like fresh pearl outside of it, it reflects. However, if we go a bit higher, it sinks. When we throw a stone into the water, it jumps out, right? But if we throw it in gently, it sinks. There is a significant critical threshold or critical angle. The test statistic is derived from the sample statistic extracted from the table used for the test.

It refers to a fierce statistical lock with the correct values, talking about bias in the sample. This is called verification statistics because it is used for verification. In other words, a sample has been extracted, and we are discussing statistics related to that sample. This is what is being verified. The p-value is the probability of obtaining a test statistic that is greater than the absolute value of the statistic obtained from the sample. You might find that you will use p-values a lot when you go to graduate school for a doctoral program, but it is not often used in descriptive statistics. You might think, Oh, this exists, but that should suffice. There is a hypothesis testing model. You know the range for hypothesis testing, right? Typically, it is 95%. These are values that you might have encountered, such as 2 - 2.5, and the t-values from the tables. So, we set the test statistic based on the sample.

I have obtained statistics regarding the surface, but we did not adhere to our traditional little unity. If the figures fall within this range, it will adopt the existing scale, and the current champion will continue to be the champion. However, if the control value for the surface exceeds this range, what will happen? The research hypothesis will be accepted if it goes outside a certain area. If it exceeds 2.57, a new research hypothesis will be accepted, while if it remains within this range, the existing scale champion will continue to be maintained. This is an example of hypothesis testing. You should keep this graph in mind and carefully observe whether the supporting statistical value falls within this range or outside of it.

You need to see both sides of the argument and the singular emotion. First, let's discuss the dual perspective. The research hypothesis is that our population is different from others. We are different from you. This is because the existing champions, like the H materials, are similar to us. We look almost like the members of TWICE. In our class, the weights are almost the same as the nine members of TWICE. That's what I mean. The friends sitting next to us might be saying, What are you talking about? That's not true. Our vocal skills are similar to those of TWICE. This is already known. Now, there are some new counterarguments. They might say, What are you talking about? That's not true. But that's exactly it. Yes, our vocal skills are similar to those of TWICE. But what does that mean? And among us, if there are 50 people, we can pick about 10 of them.

When we select and see that the skills are the same, if Lee Ha-nee is included, then that statement is correct. If we go outside, then your hypothesis until now will be proven wrong. That's how we approach it. Therefore, we need to have a contrasting hypothesis that goes against the initial one. You understand this, right? If not, there is a distance on the left and a group on the right. The left group has a sample size of 30. We have our average group, which is greater than a certain value. Should we not know this? The average height of our male students is reported to be over 175 cm. That's how it has been known. We know that the average height of male students is over 175 cm. How is that known? However, upon request, it seems otherwise. The average height of male students is said to be a packet, and that’s the report. How was this done?

Our male students are much taller than the male students in the language department. I'm sorry for constantly bringing up height, but it can't be helped. So, it is what it is. Our male students are significantly taller than those in the language department. What can be said to counter that? No, that's just the way it is. Then how about this, teacher? We have samples, right? Do you remember the textbook we have? This time, it’s on my right side. It’s been traditionally known as such. What is traditionally known? The average weight of the female students in our class is 53 kilograms. I think that has been raised a bit. It has been said that it’s 53 kilograms until now, but looking at the male students, it doesn’t seem to be the case.

It's not like that; we need to check the gold and black. Then, we should select ten female students and see if the weapon passes the test. If it comes in with the stick, it won't work, and if it enters here, it will bounce out, you know? Another thing is that the weight of the female students is smaller than that of the male students. That's not true; we are bigger. We need to consider that as well, and we have to do the checks. So, there are times when we conduct two-sided tests, and there are instances when we check the left distance and the right distance. It varies depending on the situation. In certain cases, we determine whether it's a left-sided test or a right-sided test based on the circumstances. There’s nothing difficult about it. We use the t-value of the statistic for hypothesis testing. In hypothesis testing, when we accept the research hypothesis, when do we accept it in a two-sided test? We accept it when it goes outside the critical value. If it's a left-sided test, we accept the research hypothesis under those circumstances.

When we make a selection, we calculate the sample. If this is a normal distribution and we are looking at 95%, then if the value is greater than -1 or 1.96, that would be significant. In that case, researchers would adopt it. The right-tailed test is the same as the left-tailed test; there wouldn't be any difficulty, right? Therefore, to always test this, we calculate the sample size and perform the necessary statistical tests. You know that, right? Depending on whether the value falls within the range, if it does, we reject the null hypothesis, and if it doesn't, we accept it. This is a situation where new researchers benefit. You can think of it as having conditions for hypothesis testing and whether or not to adopt a hypothesis.

The materials are referring to the current level. Until now, this has been correct, and we are adopting this. The right decision is to reject the incorrect champion. If something is correct but labeled as wrong, that is an error. Conversely, if something is wrong but labeled as correct, that is also incorrect. Do you understand? Of course, I will explain this in detail in Chapter 10 with a table. In hypothesis testing, when we choose a champion, we are considering the existing champion's errors. The threshold is referred to as the critical value. Do you see? Therefore, if the statistic related to the sample falls within this range, the champion's hypothesis is accepted; if it goes outside, the challenger’s research hypothesis is accepted. There is no difficulty, so let's proceed.

I will proceed. There is a hypothesis about the scale. We can avoid it, right? There is a champion; the champion is correct. The champion is rising, but we checked it and only obtained the black mark to adopt it. The correct decision is to provide the hypothesis accurately. However, we set the conditions too strictly. If we eliminate it, that would be wrong. We can make two decisions. The correct point is the true value that we want to extract from the cube hypothesis. The true value is coming out. The scale hypothesis has come up, and we adopted it. That means it is the correct decision, and we are using the scale hypothesis to argue. But let’s go with the existing scale hypothesis. Let's say the truth is four. Let’s say the truth is true. This time, the scale hypothesis is incorrect, but we adopted it. It means we are getting lost. You have given us the bingo hypothesis.

But it’s saying that it’s a lie. The rejection was made. It’s the right decision, but I still don’t know the true value, right? I only know that it’s not mine. I know that the existing claim is wrong, but I still don’t know the exact value. In this case, the resistance value is not known. This case is correct, but it’s a case where we don’t know the exact value. Looking at the solution, it seems easy for you, so I will proceed with it. 

The hypothesis testing range is on the right side, right? Our female students weigh 53 kg. So, we selected 10 female students and measured their weights, and they fit within this range. Then, the existing claim becomes correct. However, when checking the threshold, it shows the opposite or a higher value, which means the hypothesis is incorrect. But with this graph, it’s like this, and it’s wrong. If it turns out to be the case, we will see the actual product.

At this point, if something is rejected and it is actually incorrect, or if it differs from the actual pump, an error will occur. Let's take a look at the steps of hypothesis testing one by one. These are shown to enhance your understanding. 

In hypothesis testing, the first step involves establishing the null hypothesis. This is the content that we want to prove, and the null hypothesis states that there is no effect or no difference. For example, we might say that the average height of the boys in our class is 170 cm, which is the existing hypothesis we have been working with. However, we want to prove that the average height is not 170 cm, but rather 175 cm. This is the alternative hypothesis. 

How do we go about this? This will involve one-tailed testing. We will be testing whether the average height is greater than 175 cm. If we don't understand this, we can look at the graph on the right and say, No, that’s not the case.

No, it's between 175. If I listen to the left distance, it sounds like it's over 175, but it’s not. Right? So, this will be the left distance. Intuitively, what can I say? Our singing ability is higher than Girls' Generation, but these days, it seems like Twice is taking over, right? Something like that. It's not even comparable, and I can't hear it either. So sometimes we need to check both sides, and sometimes we need to decide whether it's the left or right side. Yousujin has to make the decision. 

This is just simple. Should I go with 90% or 95%? Do I want the confidence interval to be 99%? Therefore, I think I’ll be at 10%, and then 5% would be the next level, and 1% would be the lowest. So, 10% becomes 0.2, 5% becomes 0.05, and since it's 1%, it becomes 0.02. This is something the analyst has to decide: how much I want.

할래 유수 전 얼마로 할래? 90을 할래? 95로 할래? 아래 99를 에 따라서 요 수준 결정 되는 거니까 어려운 거 없죠. 유 수준이 작을수록 엄격하게 기준으로 수행되는 가설 검증 해요. 1% 호 퍼센트, 10% 빡빡 값이 이거예요. 자가 지겠죠? 그분 허용 거는 90, 95, 99니까 자가 gr 팝스. 자가 진단 얘기는 신뢰가 커지는 거잖아요. 요 수준이 자 것으로 엄격한 기준으로 수행한다는 얘기에요. 일반적으로 요소들은 0.05나 0.01 미만으로 한다는 얘기겠죠. 95%, 90%, 99% 한다는 얘기에요. 일반적으로 구조 퍼센트를 많이 하죠. 어려운 건 없다고 보여집니다. 연구의 목적과 중요성을 고려해서 요 수준을 결정한다라고 되어 있네요. 임계치 산출하는 거에요. 임 뷰트 산출 이것도 어려움이 절대 없죠. 95%고 정규분포라면 뭐 -1.96 유+ 주고 정도 나오는 거죠. 우리가 알고.

What level do you want to set? Do you want to set it at 90? Or at 95

There is nothing difficult at all. When it comes to a two-tailed test, it would be around 1.96, which corresponds to 95% and 1 in 4. If we consider the left tail, it would be around 6%, which becomes 5% and 95%. In this case, for a two-tailed test, we have 2.5% on each side. Therefore, 1.96 is derived from this left tail. For 95%, what happens here is 5%. So for a 90% confidence interval, it would not be 1.96, but rather what? It would be around -1.645, right? You know that because it's one-tailed. In a two-tailed test, we have 2.5% on each side. When it comes to the left tail, it is already 5%. If the right tail is considered, it would be this way, and that would be 5%, which gives us plus 1.645.

이해가 되겠죠. 언더 스탠드 8쪽만 했으니까, 예를 들어 95%면 애가 5%니까 이때는 플러스 1.6이 되겠죠. 그때는 - 이쯤 있어요. 양측 검정일 때는 2.5, 2.5니까 -2 정도 굴욕을 파실 수 있는 거예요. 95% 이때 이거 이해하면 9월 공부는 끝난 겁니다. 어렵게 하나도 없어요. 가설 채택 여부를 결정하면, 표본을 구해 가지고 표본 수치 값을 구하면 되는 거예요. 그리고 그동안 많이 한 거잖아요. 이렇게 구하는 거 많이 해보셨죠? 요구하면 되는 거예요. 구해 가지고 통계량을 구한 다음, 이를 임계치와 비교하여 연구 가설의 채택 여부를 결정하는 겁니다. 통계량을 취해가지고 검정 통계량의 t 값을 구하면 돼요. 이렇게.

The burden is that the formulas identified earlier are not at all difficult to understand. You can think of it as a technical value and a table. Instead, it's like solving a lottery problem. You can view it this way. I will also show you examples from when I was in the fourth grade, so you can look at these as well. I will keep repeating this for you.

The decision on whether to adopt a method is the same content; I am providing the same information. The Murphy's Capo comes up, and we are calculating the statistics for the sample. If it falls within this range, it means the champion is correct; if it falls outside, it means the champion is incorrect. Researchers believe it is correct.

In the case of a two-sided test, the probability of obtaining a more extreme value than the current statistic from other samples is called the p-value. This refers to the area under the curve. This is what the researchers are referring to as correct.

If the existing champion is incorrect, it would mean that extreme values are produced. Similarly, if the statistical value or research hypothesis falls within a certain range, and the t-value is around 1 or 10, if the calculated value falls within the margin of error or the confidence interval, then the existing champion is correct. However, if it falls outside, then the research hypothesis is correct. This means that if the new hypothesis proves to be incorrect, it indicates that the t-value falls within a certain range, suggesting there are no difficulties. If it passes, then the statistical value is accepted. In cases where the research hypothesis is rejected, if the previous data, such as housing prices, falls within that range, then the research hypothesis is incorrect, and the existing champion is correct. But if the t-value calculated shows an extreme value, then the research hypothesis is accepted.

The existing champion is wrong. If you understand this much, then your study for today is over, and from now on, you can just relax and observe. Let’s solve an example. If we study just one example, I think today’s learning will be complete. Through this example, we will elevate what we have studied so far. There is a type of bread called gun bread, which you probably like, right? Gun bread has a nickname: star candy. When you go to the military, there is gun bread, and inside the gun bread, there are star candies wrapped in plastic. It is said that the average number of star candies is 40. However, since some people eat more, some of their friends say they have less, while others say they have more. It can be quite annoying. So, we need to figure out whether the average of 40 star candies is accurate or not.

So after that, I randomly selected 10 samples: 37, 38, 39, 44, 14, 10, 41, 42, and 43. They say there are 40 in total. I was given the standard deviation. A sample of 10 was taken. The level of significance is alpha percent. A few things have changed. So, we will determine whether the original 40 is correct or not. Since there are 40, we will check if the new data is less than or greater than 40. Let's see how to do this from now on. 

In hypothesis testing, what is the existing hypothesis? Our null hypothesis is that we have 40 samples. It’s not a refutation, but you know, right? What do we call H zero? Up to now, we have been discussing H one, which is what has just been proposed. Our time is also defined as such: t0 refers to the current time, t1 refers to one year later, t2 refers to two years later, and t3 is now. So, t1 refers to the next.

The T2 is the following month, two months later. Last month, that refers to two months ago, especially. Let's start with the phone, the threshold, and the period area. Oh, the pool is high. Is it a selection? Since it’s 5%, this is a good state, right? The structure, that group, is below 5%, so it combines with 95%. It gives a percentage. Here, what do we have? It’s negative 1. Here, it’s plus 1.96. This is what we need to organize for the two-tailed test. Since all 40 are the same, we could catch 45, and we could also say that, right? We’re doing it with both sides. The threshold is 1.96. We check to see if it goes to -2.96. The acceptance criterion is that if the Z value falls within this range, it’s accepted; if it goes outside, it’s rejected. That’s what it indicates. You understand, right? After quitting smoking, it’s about the average of these 10.

You know, if we calculate the z-value, we can do it like this. It turns out to be 0.158. When we compute it, we get a value of about 40.1 minus 40, which gives us 0.158. This is how we narrow it down. Now, I'm not sure if there are any graphs, but let me explain. It would come out like this: as mentioned earlier, -1.96 and +1.96, and the 0.158 falls within this range. Since it is less than 1.96, it fits in here, confirming that the existing claim is correct. I think I explained that well. Our calculated t-value is 1.158. In other words, it falls within this range of -2.96 and +1. This indicates that the null hypothesis is indeed correct.

Therefore, we can confidently say that there are 40 star candies. Also, regarding the baseball, we need to use the rubber ball. For example, it weighs 500 grams. It fits inside, right? Some might argue that it seems small and light, while others might think it feels heavy. How do we resolve this? We can take about 30 samples of baseballs and see if they fit the criteria. After testing, if we calculate the 95% confidence interval, we can determine if it matches the 500 grams, and if it falls below, it’s small, and if it goes above, it’s heavier. This is how we conduct a two-tailed test. There’s no difficulty in this. We can also find this using Excel. If we have 10 numerical values, we know how to calculate the average, right? By activating the average function, we can get the result. How do we calculate the standard deviation? This might be something to consider at that time.

Is that right? Why is that? This is how it goes. I wrote it down, and there were a lot of things to exclude. I was trying to do this for the special offer. This is what I obtained under the title of prejudice. So, this is what I got. The meaning of 2 minutes zero is that it’s a bang. Do you understand? However, it seems to be a bit difficult. What’s the price of the book? On the next page, it must be someone living worldwide, right? Yes, let’s go. What do you say the price is? The price is... The value is... This is 0.63. Adding the sample mean of 40 points, it should be 102.43, right? That’s how I calculated it. The critical value is 1.9, which comes out, right? The standard deviation is coming out evenly. So, since this fits inside, -1.96 to +1 is included, so we can say that the assumption is correct. It’s not difficult, right? Until this time, we studied two-tailed test cases.

Next week, we will study the left and right concepts. Since there is a lot to cover, we will divide it into sections. Let's summarize what we studied today. There is hypothesis testing in English, which includes the significance level and critical value, as well as the test statistic. Please take a look at the contents thoroughly. My cat is currently meowing because it's hungry. I encourage you all to read through the terms related to hypothesis testing; I believe you will fully understand the material we covered today. Thank you for your hard work!

[경영통계학] 가설검정의 개념

주식삼촌 최정일

Hypothesis testing involves two main types of hypotheses: the null hypothesis, which challenges existing theories, and the alternative hypothesis, which suggests that the null hypothesis is not valid.

The testing process is based on the assumption that the null hypothesis is correct until proven otherwise, with the significance level set to less than 0.

Extreme statistical values may lead to the adoption of the research hypothesis, and the concept of a confidence interval helps determine the accuracy of the hypotheses.

Understanding Alpha and Hypothesis Testing

Alpha values, representing the acceptable error rate, are typically 10%, 5%, or 1%, corresponding to 0.

Hypothesis testing involves selecting a sample to determine if it falls within the acceptable range, with an extreme sample statistic indicating a potential rejection of the hypothesis.

The p-value is used to assess the probability of obtaining a test statistic greater than the sample's absolute value, playing a crucial role in hypothesis testing.

The surface statistics will determine whether to maintain the existing scale or accept a new research hypothesis.

57, a new hypothesis will be accepted; otherwise, the current champion will remain.

The discussion highlights the dual perspective of the research hypothesis, emphasizing the similarity of the population and the need for contrasting hypotheses to challenge the initial assumption.

The process involves selecting female students to test a weapon's effectiveness, considering their weight compared to male students.

Hypothesis testing is conducted using t-values and critical values to determine whether to accept or reject the null hypothesis.

The speaker emphasizes the importance of accurate decision-making and setting appropriate conditions for hypothesis testing.

Hypothesis testing involves establishing a null hypothesis, which states there is no effect or difference, and an alternative hypothesis that proposes a different value.

The testing can be one-tailed, focusing on whether the average is greater than a certain value.

The significance level, such as 90%, 95%, or 99%, is determined by the analyst and plays a crucial role in the rigor of the hypothesis testing.

Understanding Two-Tailed Tests and Confidence Intervals

Two-tailed tests use a critical value of around 1.

96 for a 95% confidence interval, with 2.

For a 90% confidence interval, the critical value is approximately -1.

The p-value in a two-sided test represents the probability of obtaining a more extreme value, and the decision to accept or reject the research hypothesis is based on whether the calculated t-value falls within the margin of error or confidence interval.

Hypothesis Testing with Gun Bread and Star Candies

The example focuses on gun bread, which has an average of 40 star candies, causing varying perceptions among people.

A sample of 10 was taken to determine if the average is accurate, using hypothesis testing with a null hypothesis of 40.

The calculated t-value and z-value both fell within the acceptance range, confirming that the null hypothesis is correct.

Conducting a Two-Tailed Test for Baseballs

There are 40 star candies, and the rubber baseball weighs 500 grams.

Taking 30 samples of baseballs can help determine if they fit the criteria, and calculating the 95% confidence interval will indicate if they are small or heavy.

The average and standard deviation can be calculated using Excel, and the critical value of 1.

9 suggests that the assumption is correct.