Generate a 5-digit number 100 times. Record the number of even numbers in
5-digit numbers. Fit a binomial distribution to this data set. Test the
hypothesis whether the binomial distribution gives a satisfactory fit to the
data.
Answers
Explanation:
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In this type of hypothesis test, you determine whether the data “fit” a particular distribution or not. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.
The test statistic for a goodness-of-fit test is:
∑
k
(
O
−
E
)
2
E
where:
O = observed values (data)
E = expected values (from theory)
k = the number of different data cells or categories
The observed values are the data values and the expected values are the values you would expect to get if the null hypothesis were true. There are n terms of the form
(
O
−
E
)
2
E
.
The number of degrees of freedom is df = (number of categories – 1).
The goodness-of-fit test is almost always right-tailed. If the observed values and the corresponding expected values are not close to each other, then the test statistic can get very large and will be way out in the right tail of the chi-square curve.
Note: The expected value for each cell needs to be at least five in order for you to use this test.
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