There are five main entrances to shopping mall and the mall manager would like to know if the entrances are equally utilized. To investigate, 400 people were observed entering the Mall. The number using each entrance is reported below.Entrance FrequencyMain Entrance 96Petchkasim Road 121Garage 90North 93 State the alternative hypothesis χ2 Critical Value =______ χ2 test statistic =______ Decision? Conclusion?A. The observed data fit the expected entrance usage. B. Reject H0.C. 7.067D. There is sufficient evidence to conclude that entrance usage does not fit the expectation.E. There is insufficient evidence to conclude that entrance usage does not fit the expectation.F. 7.815G. 6.06H. Do Not Reject H0.I. 12.822J. 9.488K. The observed data do not fit the expected entrance usage.
Question
There are five main entrances to shopping mall and the mall manager would like to know if the entrances are equally utilized. To investigate, 400 people were observed entering the Mall. The number using each entrance is reported below.Entrance FrequencyMain Entrance 96Petchkasim Road 121Garage 90North 93 State the alternative hypothesis χ2 Critical Value =______ χ2 test statistic =______ Decision? Conclusion?A. The observed data fit the expected entrance usage. B. Reject H0.C. 7.067D. There is sufficient evidence to conclude that entrance usage does not fit the expectation.E. There is insufficient evidence to conclude that entrance usage does not fit the expectation.F. 7.815G. 6.06H. Do Not Reject H0.I. 12.822J. 9.488K. The observed data do not fit the expected entrance usage.
Solution
The question is asking for a chi-square test for goodness of fit.
Step 1: State the Hypotheses The null hypothesis states that the observed frequencies for a categorical variable match a specified distribution. For this case, the null hypothesis is that the entrances are equally utilized. The alternative hypothesis is that the entrances are not equally utilized.
Step 2: Find the Expected Frequencies The expected frequency for each category is the total number of observations divided by the number of categories. Here, we have 400 people and 5 entrances, so the expected frequency for each entrance is 400/5 = 80.
Step 3: Calculate the Test Statistic The chi-square test statistic is calculated as the sum of the squared difference between observed and expected frequencies divided by the expected frequency, for all categories.
χ2 = Σ [ (Observed-Expected)^2 / Expected ]
For Main Entrance: (96-80)^2 / 80 = 3.2 For Petchkasim Road: (121-80)^2 / 80 = 20.8125 For Garage: (90-80)^2 / 80 = 1.25 For North: (93-80)^2 / 80 = 2.0875
Adding these up, χ2 = 27.35
Step 4: Find the Critical Value The critical value for a chi-square distribution can be found in a chi-square distribution table. The degrees of freedom for this test is the number of categories minus 1, which is 5-1=4. For a significance level of 0.05, the critical value is 9.488.
Step 5: Decision Since the test statistic (27.35) is greater than the critical value (9.488), we reject the null hypothesis.
Step 6: Conclusion There is sufficient evidence to conclude that entrance usage does not fit the expectation. The observed data do not fit the expected entrance usage.
Similar Questions
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