What Are The Parameters That Indicate The Need For Intubation – If you Google “what is a p-value”, the first result on the page is the Wikipedia definition:
“A p-value is the probability of obtaining a test result that is at least as severe as the actually observed result, assuming the null hypothesis is true.” – Wikipedia
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Well… luckily we know this definition is correct; the bad thing is that it’s too precise to understand.
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Santa’s Cookie Shop is selling their most popular product – Gingerbread Cookies! Santa is very proud of his cookies. He believes his products are the best tasting in the world. Also, Santa says that the average weight (μ) of each product (a bag of gingerbread) is 500 grams.
It’s the 21st century. Santa already has his own factory and automated machines to help make the cookies. As you know, there is no perfect machine and production process, so there are differences between different bags of cookies. We know that the cookie bags are normally distributed with a standard deviation (σ) equal to 30g.
So, if Satna’s statement is correct (average weight of a bag of biscuits = 500 grams), we can expect the following distribution of biscuits per bag:
But, as an avid customer who really loves gingerbread, I wonder…is the average weight of a bag of biscuits really equal to 500 grams? What if Santa cheats customers and gives us cookies that are less than 500 grams? How do we verify Santa’s words?
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To run a hypothesis test, first, let’s set up the null hypothesis (H0) and the alternative hypothesis (H1). As a reasonable person, one should not suspect others without any evidence. Therefore, we assume that Santa Claus is honest (H0) about his business. If we want to check if his cookies are less than 500g, we need to collect data and have enough evidence to support our hypothesis (H1). So…we build the hypothesis statement as follows:
H0: Average weight of a bag of biscuits (μ) = 500g H1: Average weight of a bag of biscuits (μ) < 500g
Since we don’t know exactly what our population distribution will be, I’ve used dashed lines to represent possible distributions. If Santa’s claim is true, we can expect to distribute a bag of cookies with an average weight of 500 grams (left). However, if Santa’s statement is incorrect and the average cookie weight is less than 500 grams, the population distribution should look different (anything in the correct number).
Cool! The problem statement is set. So now, the next question is: How do we test our hypothesis statements?
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Can we weigh all the cookie bags so we know the exact population distribution? Hmmm… obviously there’s no way we’ll collect all the cookies (populations) that Santa’s Cookie Shop produces! ! ! So… what do we do?
In inferential statistics we are interested in population parameters (population characteristics). However, it is almost impossible to collect all the data of the whole population to calculate the parameters. Therefore, we sample from the entire population to obtain sample data. We then compute statistics (characteristics of the sample) from the sample data as our estimators to help us infer unknown population parameters. (As shown below)
Examples of parameters and statistics: – Parameters: population mean (μ), population standard deviation (σ) … – statistics: sample mean (x̄), sample standard deviation (s) …
Testing our hypothesis statements is also an inferential statistical task. The process is the same as above. But right now, we’re not interested in the unknown parameter; instead, we’re asking “Can we reject the null hypothesis?” We’re interested in it.
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How to answer this question? Using the same approach – we compute statistics from the sample data to predict the answer to this question. The statistics used here are called test statistics.
Great! So now, we know that we need to collect sample data and calculate test statistics to test the hypothesis statement.
But… let’s pause for a moment. Before getting into the testing part, let me quickly review the concept of sampling distributions to make sure we are on the same page.
Let’s take one of these statistics – the sample mean (x̄) – as an example. If we sample multiple times from the population, we can get several sample data sets (from sample 1 to sample m). Then, if we calculate the sample mean (x̄) from each sample dataset, we can get m data points for the sample mean (x̄). Using these data points, we can plot the distribution of the sample mean (x̄). Since this distribution comes from the sample statistic, we call this distribution the sampling distribution of the sample mean (x̄).
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The same idea applies to other statistics as well. For example, if we compute the test statistic from each sample data set, we can obtain the sampling distribution of the test statistic.
A sampling distribution is similar to all other distributions in that it shows the likelihood (probability) of a statistical value occurring if we sampled multiple times from the population.
The first thing we need is to have a sample dataset. So, I went to Santa’s cookie shop and randomly picked 25 bags of cookies (n) as our sample data. Also, I calculated the average weight (x̄) of this sample to be 485g.
The first part of the test is comparing our sample statistic to the null hypothesis so we know how far our sample statistic is from the expected value.
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To do this, we first assume the null hypothesis to be true. What does it mean? This means that, in our example, we assume that the population distribution of a bag of biscuits is indeed equal to 500 grams. And if this statement is true, then according to the central limit theorem, if we sample from this population many times, we may have a sampling distribution of the sample mean (x̄), as shown in the figure below (sample mean = 500g).
Definition of p-value: “The p-value is the probability of obtaining a test result that is at least as extreme as the actually observed result, assuming the null hypothesis is true.” – Wikipedia
So now, if the null hypothesis is true, we can easily see that our sample mean is 15g (485–500 = -15) smaller than the predicted mean (500g).
Hmm… But “15g” is just a number and doesn’t help us explain the meaning much. Also, if we want to calculate the probability under the curve, it’s inefficient to calculate it alone (imagine having many distributions, each with its own mean and standard deviation… you really shouldn’t be calculating probabilities most of the time …)
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So, what should we do? We normalize our values so that the mean of the distribution is always zero. The benefit of normalization is that the statistician has created a table containing the area under each normalized value. So we don’t need to calculate the area separately. All we need to do is validate our data.
How to authenticate? In our case we use z-score to transform our data. In our case, the z-score is the test statistic.
The figure below shows the sampling distribution of the test statistic (z-score). If our sample data happens to be equal to the null hypothesis (population mean = 500g, sample mean = 500g), we should make the test statistic equal to 0. In our case, our sample mean equals 485g, which gives us a test statistic equal to -2.5 (-2.5 is the test result we observed from our sample data). This shows that our sample data is 2.5 standard errors higher than the predicted value.
Here, what I want to say is that the test statistics are selected according to different situations. You may have heard of different types of statistical tests such as z-test, t-test, chi-square test…why do we need different types of tests?
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Since we may need to test different types of data (categorical? quantitative?), we may have different testing goals (test mean? ratio?), the data we have may have a different distribution, our data may have limited characteristics… .. Therefore, how to choose a suitable test method is another crucial task.
In this case, since we are interested in testing the mean, I will assume that our population data is normally distributed with a known population standard deviation (σ). In our case, we choose the z-test for this case. ** If you are interested in when to use each statistical test, see Estimates for Different Types of Statistical Tests. **
Ok, now we have test statistics. We know how far our test statistic is from the expected value when the null hypothesis is true. So, what we really want to know is: if the null hypothesis is true, what is the likelihood (probability) that we get this data sample?
To answer this question, we need to calculate probabilities. As you know, the probability between one point and another is the area under our sampling distribution curve between those two points.
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So here, we don’t calculate the probability of a particular point; instead, we calculate the probability
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