Question

You are conducting a multinomial Goodness of Fit hypothesis test for the claim that the 4 categories occur with the following frequencies:

HoHo : pA=0.4pA=0.4;  pB=0.25pB=0.25;  pC=0.25pC=0.25;  pD=0.1pD=0.1

Complete the table. Report all answers accurate to three decimal places.

Category	Observed Frequency	Expected Frequency
A	34	36
B	11	22.5
C	15	22.5
D	30	9

What is the chi-square test-statistic for this data?
χ2=χ2= 57.489

What is the P-Value?
P-Value = 0

For significance level alpha 0.025, what would be the conclusion of this hypothesis test?

• Reject the Null Hypothesis
• Fail to reject the Null Hypothesis

Explanation of answer is below!

We are conducting a goodness-of-fit test. How many degrees of freedom do we have? That's the number of categories minus one. We have 4 categories, so we have 3 degrees of freedom.

We now want to input the observed frequency into L1 and the expected frequency into L2. We need to first find the expected frequencies. How many subjects were observed? Let's add up the values from each category: 34+11+15+30=9034+11+15+30=90.

How many subjects to we expect in each category?

• Since we expect 0.4 of the 90 observations in category A, we would expect 90⋅0.4=3690⋅0.4=36 in category A.
• Since we expect 0.25 of the 90 observations in category B, we would expect 90⋅0.25=22.590⋅0.25=22.5 in category B.
• Since we expect 0.25 of the 90 observations in category C, we would expect 90⋅0.25=22.590⋅0.25=22.5 in category C.
• Since we expect 0.1 of the 90 observations in category D, we would expect 90⋅0.1=990⋅0.1=9 in category D.

Input these values into L2.

Now you can run the χ2χ2 GOF test in your calculator (found in STAT >> TESTS in your calculator). Select L1 as the Observed, L2 as the Expected, and 3 degrees of freedom. The calculator should output the test statistic and p-value.

If the p-value is less than the significance, then we reject the null. Otherwise, we fail to reject the null.

Answer 1

The detailed solution is given below.