Statistics Chi Square test of normally distributed data

In summary, the conversation revolves around using estimators and the Chi-squared test to determine if a data set with a hypothesized normal distribution is sustained. The speaker asks for guidance on using estimators when the data is continuously distributed in bins. The response explains the calculation of bin probabilities using the standard normal distribution and suggests using normal tables or a computer package for this calculation.
  • #1
Liesl
2
0

Homework Statement


The following data is hypothesized to possesses a normal distribution, is that hypothesis sustained by a Chi-square test at .05 significance:

<66...4
67-68...24
68-70...35
70-72...15
72-74...8
>74...4


The Attempt at a Solution



1. I know that I should calculate the expected values for each bin for a normally distributed data set using the equation f(x)=2.66*e^-[(x-μ)^2/(2σ^2)] To do this, I'lll need to calculate μ and s from the data using estimators.

2. Then I can simply plug those values into the Chi Squared equation and see if the resulting value is less than the X^2 value from the table. (I will have to make sure np > 5 for all bins, and subtract 2 from the number of final bins to get the degrees of freedom).

My question is, how do is use estimators when the data is continuously distributed in bins as it is in the problem? Thank you.
 
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  • #2
I would try to minimize that Chi-squared value in a fit to get mean and standard deviation (and Chi2) at the same time.
 
  • #3
Liesl said:

Homework Statement


The following data is hypothesized to possesses a normal distribution, is that hypothesis sustained by a Chi-square test at .05 significance:

<66...4
67-68...24
68-70...35
70-72...15
72-74...8
>74...4


The Attempt at a Solution



1. I know that I should calculate the expected values for each bin for a normally distributed data set using the equation f(x)=2.66*e^-[(x-μ)^2/(2σ^2)] To do this, I'lll need to calculate μ and s from the data using estimators.

2. Then I can simply plug those values into the Chi Squared equation and see if the resulting value is less than the X^2 value from the table. (I will have to make sure np > 5 for all bins, and subtract 2 from the number of final bins to get the degrees of freedom).

My question is, how do is use estimators when the data is continuously distributed in bins as it is in the problem? Thank you.

You calculate the bin probabilities in the usual way: ##P\{ a < X < b\} = P\{ (a - \mu)/\sigma < Z < (b - \mu)/\sigma\}##, where ##Z## is a standard normal random variable (mean= 0, variance = 1). You need to use normal tables or a computer package, or something similar.
 

1. What is a Chi Square test and how is it used in statistics?

A Chi Square test is a statistical test used to determine whether there is a significant difference between observed and expected values in a categorical data set. It is used to analyze data that is organized into categories or groups.

2. What is the assumption for using a Chi Square test on normally distributed data?

The assumption for using a Chi Square test on normally distributed data is that the data is independent and randomly sampled from a population, and the expected frequencies in each group should be at least 5.

3. How do you interpret the results of a Chi Square test?

The results of a Chi Square test are typically presented in the form of a p-value. If the p-value is less than the chosen significance level (usually 0.05), then we reject the null hypothesis and conclude that there is a significant difference between the observed and expected values. If the p-value is greater than the significance level, we fail to reject the null hypothesis and conclude that there is no significant difference.

4. Can a Chi Square test be used for continuous data?

No, a Chi Square test should only be used for categorical data. For continuous data, other statistical tests such as t-tests or ANOVA should be used.

5. What is the difference between a Chi Square test and a t-test?

A Chi Square test is used to analyze categorical data, while a t-test is used to compare the means of two continuous variables. Additionally, a Chi Square test is non-parametric, meaning it does not require the data to be normally distributed, while a t-test is parametric and assumes the data is normally distributed.

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