Stats chapter 12 notes PDF

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STATS CHAPTER 12 NOTES Definitions: ● Chi-square distribution ○ A distribution obtained from the multiplying the ratio of sample variance to population variance by the degrees of freedom when random samples are selected from a normally distributed population ● Contingency Table ○ Data arranged in table form for the chi-square independence test ● Expected Frequency ○ The frequencies obtained by calculation. ● Goodness-of-fit Test ○ A test to see if a sample comes from a population with the given distribution. ● Independence Test ○ A test to see if the row and column variables are independent. ● Observed Frequency ○ The frequencies obtained by observation. These are the sample frequencies. Properties of Chi Square: ● ● ● ●

Chi-square is non-negative. Is the ratio of two non-negative values, therefore must be non-negative itself. Chi-square is non-symmetric. There are many different chi-square distributions, one for each degree of freedom. The degrees of freedom when working with a single population variance is n-1.

Chi Square Probabilities: Since the chi-square distribution isn't symmetric, the method for looking up left-tail values is different from the method for looking up right tail values. ● ● ●

Area to the right - just use the area given. Area to the left - the table requires the area to the right, so subtract the given area from one and look this area up in the table. Area in both tails - divide the area by two. Look up this area for the right critical value and one minus this area for the left critical value.

Goodness-of-Fit Test: The test statistic has a chi-square distribution when the following assumptions are met: ●

The data are obtained from a random sample

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The expected frequency of each category must be at least 5. This goes back to the requirement that the data be normally distributed. You're simulating a multinomial experiment (using a discrete distribution) with the goodness-of-fit test (and a continuous distribution), and if each expected frequency is at least five then you can use the normal distribution to approximate (much like the binomial). If the expected

The following are properties of the goodness-of-fit test: ● ● ● ● ●

The data are the observed frequencies. This means that there is only one data value for each category. Therefore, ... The degrees of freedom is one less than the number of categories, not one less than the sample size. It is always a right tail test. It has a chi-square distribution. The value of the test statistic doesn't change if the order of the categories is switched....