Chi-Square (11.1)

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the following school year.  She wants to know if each type of food will be equally popular so she can start ordering supplies making other plans.  To find out, she selects a random sample of 100 students and asks them, "Which type of food do you prefer: Asian food, Mexican food, pizza, or hamburgers?"  Here are the data in the table provided.  An appropriate null hypothesis to test whether the food choices are equally popular is:

• H₀: μ = 25, where μ = the mean number of students that prefer each type of food.
• H₀: p = 0.25, where p = the proportion of all students who prefer Asian food.
• H₀: η_A = η_M = η_P = η_H = 0.25, where η_A is the number of students in the school who would choose Asian food, and so on.
• H₀: p_A = p_M = p_P = p_H = 0.25, where η_A is the proportion of students in the sample who would choose Asian food, and so on.

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.4

CCSS.HSS.ID.B.5

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the following school year.  She wants to know if each type of food will be equally popular so she can start ordering supplies making other plans.  To find out, she selects a random sample of 100 students and asks them, "Which type of food do you prefer: Asian food, Mexican food, pizza, or hamburgers?"  Here are the data in the table provided.  The chi-squared statistic is:

• (18-25)²/25 + (22-25)²/25 + (39-25)²/25 + (21-25)²/25
• (25-18)²/18 + (25-22)²/22 + (25-39)²/39 + (25-21)²/21
• (18-25)/25 + (22-25)/25 + (39-25)/25 + (21-25)/25
• (0.18-0.25)²/0.25 + (0.22-0.25)²/0.25 + (0.39-0.25)²/0.25 + (0.21-0.25)²/0.25

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.3

CCSS.HSS.IC.B.4

CCSS.HSS.ID.A.2

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the following school year.  She wants to know if each type of food will be equally popular so she can start ordering supplies making other plans.  To find out, she selects a random sample of 100 students and asks them, "Which type of food do you prefer: Asian food, Mexican food, pizza, or hamburgers?"  Here are the data in the table provided.  The P-value for a chi-square test for goodness of fit is 0.0129.  Which is the following is the most appropriate conclusion?

• Because 0.0129 is less than α = 0.05, reject H₀.  There is convincing evidence that the food choices are equally popular.
• Because 0.0129 is less than α = 0.05, reject H₀.  There is not convincing evidence that the food choices are equally popular.
• Because 0.0129 is less than α = 0.05, reject H₀.  There is convincing evidence that the food choices are not equally popular.
• Because 0.0129 is less than α = 0.05, fail to reject H₀.  There is convincing evidence that the food choices are equally popular.

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.3

CCSS.HSS.ID.B.5

Which of the following is false?

• A chi-squared distribution with k degrees of freedom is more right-skewed than a chi-square distribution with k+1 degrees of freedom.
• A chi-square distribution never takes negative values.
• The degrees of freedom for chi-square test is determined by sample size.
• The area under a chi-square density curve is always equal to 1.

A chi-square test is used to test whether a 0 to 9 spinner is "fair" (that is, the outcomes are all equally likely).  The spinner is spun 100 times, and the results are recorded.  The degrees of freedom for the test will be

• 8
• 9
• 10
• 99

Recent revenue shortfalls in a midwestern state led to a reduction in the state budget for higher education.  To offset the reduction, the largest state university proposed a 25% tuition increase.  It was determined that such an increase was needed simply to compensate for the lost support from the state.  Separate random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university's budget at current levels.  The results are in the table provided.  Why hypotheses would be appropriate for performing a chi-square test?

• The null hypothesis is that the closer students get to graduation, the less likely they are to be opposed to tuition increases.  The alternative is that how close students are to graduation makes no difference in their opinion.
• The null hypothesis is that the mean number of students who are strongly are opposed is the same for each of the 4 years.  The alternative is that the mean is different for at least 2 of the 4 years.
• The null hypothesis is that the distribution of student opinion about the proposed tuition increase is the same for each of the 4 years at the university.  The alternative is that the distribution is different for at least 2 of the 4 years.
• The null hypothesis is that year in school and student opinion about the tuition increase in the sample are independent.  The alternative is that these variables are dependent.

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.5

Recent revenue shortfalls in a midwestern state led to a reduction in the state budget for higher education.  To offset the reduction, the largest state university proposed a 25% tuition increase.  It was determined that such an increase was needed simply to compensate for the lost support from the state.  Separate random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university's budget at current levels.  The results are in the table provided.  The condition for carrying out the chi-square test are: 1.  Independent random samples from the populations of interest.  2.  All expected counts are at least 5.  3.  The population sizes are at least 10 times the sample sizes.  Which of the conditions is (are) satisfied in this case?

• 1 only

• 1 and 2 only

• 2 and 3 only

• 1, 2, and 3

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.3