Exam 14: Analysis of Categorical Data

For a chi-square distributed random variable with 10 degrees of freedom and a level of significance of 0.025, the chi-square table value is 20.4831. The computed value of the test statistics is 16.857. This will lead us to reject the null hypothesis.

Consumer panel preferences for three proposed fast-food restaurants are as follows: Use 0.05 level of significance and test to see if there is a preference among the three restaurants.

The chi-square technique is a statistical technique for testing the alleged independence of two qualitative variables, making inferences about the relative sizes of more than two population proportions, and conducting goodness-of-fit tests to assess the plausibility that sample data come from a population that conforms to a specified probability distribution.

The total degrees of freedom for an r * c (fixed-column) contingency table are rc + 1.

In applying the chi-square goodness-of-fit test, the rule of thumb for all expected frequencies is that each expected frequency is five or more.

Whenever the expected frequency of a cell is less than five, one remedy for this condition is to increase the size of the sample.

In a typical contingency table test, we calculate each expected cell frequency as the product of row total times column total times sample size.

A left-tailed area in the chi-square distribution equals 0.10. For 5 degrees of freedom the table value equals 9.23635.

A right-tailed area in the chi-square distribution equals 0.05. For 6 degrees of freedom the table value equals 12.5916.

Five brands of orange juice are displayed side by side in several supermarkets in Vancouver. It was noted that in 1 day, 180 customers purchased orange juice. Of these, 30 picked Brand A, 40 picked Brand B, 25 picked Brand C, 35 picked Brand D, and 50 picked Brand E. Can you conclude at the 5% significance level that there is a preferred brand of orange juice in Vancouver? Justify your answer.

In a typical chi-square test of independence, we calculate each expected cell frequency ( ) as the product of row total ( ) times column total ( ) times sample size (n).

Numbers in a contingency table show the frequency of occurrence of all possible combinations of categories.

Ink Pen Colours Narrative A national survey stated that 30% of the population prefer to use a pen with black ink, 30% prefer blue ink, 25% prefer red ink, and 15% prefer some other colour. A statistics professor took a random sample of 80 students and asked them to state their ink colour preference. The following data was recorded: -A take-out pizza palace offers four different pizzas. The manager believes that 45% of the single pizza customers prefer pepperoni pizza, 35% prefer the combination pizza, 15% prefer the taco pizza, and 5% prefer the vegetarian pizza. To check his belief, the manager takes a random sample of 100 single pizza orders and records the following information: Do the sample data present sufficient evidence to support the manager's belief? Test the appropriate hypotheses at the 10% significance level. Justify your conclusion.

A survey in Quebec claims that, among consumers, the grocery store chains of Provigo, IGA, and Metro are equally preferable. To check this claim, a random sample of 150 consumers was selected and the following data recorded: Test the appropriate hypotheses at the 5% significance level.

How many columns does the table associated with a contingency table test with 4 rows and 30 degrees of freedom have?

A contingency table classifies data according to two or more categories associated with each of two qualitative variables that are statistically independent of one another.

The number of Canadians who visit fast-food restaurants regularly has grown steadily over the past decade. For this reason, marketing experts are interested in the demographics of fast-food customers. Is a customer's preference for a fast-food chain affected by the age of the customer? If so, advertising might need to target a particular age group. Suppose a random sample of 400 fast-food customers aged 16 and older was selected, and their favourite fast-food restaurants along with their age groups were recorded, as shown in the table: Use an appropriate method to determine whether or not a customer's fast-food preference is dependent on age. Write a short paragraph presenting your statistical conclusions and their practical implications for marketing experts.

In a chi-square test of independence with 6 degrees of freedom and a level of significance of 0.05, the critical value from the chi-square table is 12.5916. The computed value of the test statistics is 11.264. This will lead us to reject the null hypothesis.

Whenever the expected frequency of a cell is less than five, one remedy for this condition is to decrease the size of the sample.

In a goodness-of-fit test, suppose that the value of the test statistic is 13.08 and the number of degrees of freedom is 6. At the 5% significance level, what may one conclude about the null hypothesis?