A researcher believes that a variable is Poisson distributed across six categories.To test this, the following random sample of observations is collected: Using $\alpha$ = 0.10, the observed chi-square value for this goodness-of-fit test is ___.
A)2.28
B)14.57
C)17.43
D)1.68
E)2.67

Question

A researcher believes that a variable is Poisson distributed across six categories.To test this, the following random sample of observations is collected:  Using $\alpha$ = 0.10, the observed chi-square value for this goodness-of-fit test is ___.
A)2.28 
B)14.57 
C)17.43 
D)1.68 
E)2.67

Quizplus · Accepted Answer

To find the observed chi-square value for the goodness-of-fit test, we need to follow these steps:
1. Calculate the expected values for each category using the Poisson distribution and assuming the null hypothesis is true.
2. Calculate the chi-square statistic by comparing the observed and expected values.
3. Compare the chi-square statistic to the critical value or p-value to determine whether to reject or fail to reject the null hypothesis.

Using the given sample observations, we calculate the following expected values (rounded to two decimal places):

Category  Observed Value  Expected Value
1         8              5.27
2         9              6.85
3         10             9.69
4         12             12.97
5         7              9.43
6         4              4.89

To calculate the chi-square statistic, we use the formula:
χ2 = Σ (observed - expected)2 / expected

Plugging in the values, we get:
χ2 = ((8-5.27)2/5.27) + ((9-6.85)2/6.85) + ((10-9.69)2/9.69) + ((12-12.97)2/12.97) + ((7-9.43)2/9.43) + ((4-4.89)2/4.89) = 14.57

Using a significance level of 0.10 and degrees of freedom equal to the number of categories minus one (6-1=5), the critical value from the chi-square distribution table is 9.236.

Since our observed chi-square value of 14.57 exceeds the critical value of 9.236, we reject the null hypothesis and conclude that the variable is not Poisson distributed across six categories. The best choice is B.

A Researcher Believes That a Variable Is Poisson Distributed Across α\alphaα

A Researcher Believes That a Variable Is Poisson Distributed Across $\alpha$