Explain the Chi – Square test.

Marvin Mccormick 2021-03-01 Answered
Explain the Chi – Square test.
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au4gsf
Answered 2021-03-02 Author has 95 answers

Step 1
Introduction:
The chi-square test can be used to test the in the following scenarios:
To test the Goodness of fit of the variables when their expected and observed frequencies are given.
To test the independence of the categorical variables by making it into a contingency table.
To test the significance of the single variance with the given variance.
1). Goodness of fit:
Goodness of fit test is applied to check how well the sample data obtained fits the distribution of the selected population. It can also be viewed as whether the frequency distribution fits the given pattern. Most commonly used test to check the goodness of fit is the chi-square test.
There are two values involved are observed and the expected values. The observed value represents the frequency of particular category in the sample and the expected value is obtained from the given distribution. Moreover, it summarizes the difference between the expected and observed values of the given data.
The hypotheses are stated as given below:
Null hypothesis: Data comes from the specified distribution.
Alternative hypothesis: Data does not come from the specified distribution.
The chi-square test statistic is calculated using the formula given below:
x2=(OiEi)2Ei
where,
Oi-Represents the observed values
Ei-Represents the expected values
Step 2
2). Test for independence:
In test for independence, we test whether there is an association between the categorical variables.
Null hypothesis: There is no association between the two categorical variables.
Alternative hypothesis: There is no association between the two categorical variables.
The chi-square test statistic is calculated using the formula given below:
x2=(OijEij)2Eij
where,
Oij-Represents the observed values
in ithcolumn and jth row
Eij-Represents the expected values
in ithcolumn and jth row
Step 3
3). Test for single variance:
Here, the chi-square test is used to compare the single sample variance with the population variance.
Null hypothesis: The given sample variance is equal to the population variance.
Alternative hypothesis: The given sample variance is not equal (less or greater) to the population variance.
The chi-square test statistic is calculated using the formula given below:
x2=(n1)s2σ2
where,
n-Sample size
s2-Sample variance
σ2-Population variance
Step 4
Examples for some research questions based on chi square tests:
1.Is there significant association between blood pressure and exercise level?
2.Whether the class has less variation in statistics marks than the other past statistics classes marks?
3.To check whether health insurance benefits vary by the size of the company.
4.To test the claim that the subject distribution of books in the library fits the distribution of books checked out by students.
5.The level of education and the amount of proceeded foods in an individual’s diet are independent or not.

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New questions

I recently have this question:
I have a bag of toys. 10% of the toys are balls. 10% of the toys are blue.
If I draw one toy at random, what're the odds I'll draw a blue ball?
One person provided an answer immediately and others suggested that more details were required before an answer could even be considered. But, there was a reason I asked this question the way that I did.
I was thinking about probabilities and I was coming up with a way to ask a more complicated question on math.stackexchange.com. I needed a basic example so I came up with the toys problem I posted here.
I wanted to run it by a friend of mine and I started by asking the above question the same way. When I thought of the problem, it seemed very clear to me that the question was "what is P ( b l u e b a l l )." I thought the calculation was generally accepted to be
P ( b l u e b a l l ) = P ( b l u e ) P ( b a l l )
When I asked my friend, he said, "it's impossible to know without more information." I was baffled because I thought this is what one would call "a priori probability."
I remember taking statistics tests in high school with questions like "if you roll two dice, what're the odds of rolling a 7," "what is the probability of flipping a coin 3 times and getting three heads," or "if you discard one card from the top of the deck, what is the probability that the next card is an ace?"
Then, I met math.stackexchange.com and found that people tend to talk about "fair dice," "fair coins," and "standard decks." I always thought that was pedantic so I tested my theory with the question above and it appears you really need to specify that "the toys are randomly painted blue."
It's clear now that I don't know how to ask a question about probability.
Why do you need to specify that a coin is fair?
Why would a problem like this be "unsolvable?"
If this isn't an example of a priori probability, can you give one or explain why?
Why doesn't the Principle of Indifference allow you to assume that the toys were randomly painted blue?
Why is it that on math tests, you don't have to specify that the coin is fair or ideal but in real life you do?
Why doesn't anybody at the craps table ask, "are these dice fair?"
If this were a casino game that paid out 100 to 1, would you play?
This comment has continued being relevant so I'll put it in the post:
Here's a probability question I found online on a math education site: "A city survey found that 47% of teenagers have a part time job. The same survey found that 78% plan to attend college. If a teenager is chosen at random, what is the probability that the teenager has a part time job and plans to attend college?" If that was on your test, would you answer "none of the above" because you know the coincident rate between part time job holders and kids with college aspirations is probably not negligible or would you answer, "about 37%?"