Which of the following statements about the chi-square (χ²) distribution is true?

• The density curve of the χ² distribution is symmetric.

• As the number of degrees of freedom increases, the mean of the χ² distribution increases.

• The mean of any χ² distribution is less than the median.

• The density curve of the χ² distribution is uniform.

From experience, the owner of an ice-cream shop has found that 60% of all sales of ice-cream cone are for one-scoop cones, 30% are for two-scoop cones, and the remaining 10% are for three-scoop cones. Recently, the shop added frozen-yogurt cones to its menu. A random sample of 250 frozen yogurt-cone sales revealed the distribution in the table. The owner wishes to know whether the pattern of sales for frozen-yogurt cones differs from that of ice-cream cones. Let p₁, p₂, and p₃ be the proportions of sales of one-scoop, two-scoop, and three-scoop frozen yogurt cones, respectively. Which one of the following pairs of null and alternative hypotheses are appropriate for this test?

• H₀: The distribution of the variable "number of scoops" is the same for frozen yogurt cones as it is for ice cream cones.

  H_a: The distribution of the variable "number of scoops" is different for frozen yogurt cones as it is for ice cream cones.

• H₀: The observed counts for "number of scoops" are the same as the expected counts.

  H_a: At least one of the observed counts for "number of scoops" is different from the corresponding expected count.

• H₀: Ice-cream cone sales and frozen yogurt cone sales are independent

  H_a: Ice-cream cone sales and frozen yogurt cone sales are dependent

From experience, the owner of an ice-cream shop has found that 60% of all sales of ice-cream cones are for one-scoop cones, 30% are for two-scoop cones, and the remaining 10% are for three-scoop cones. Recently, the shop added frozen-yogurt cones to its menu. A random sample of 250 sales of frozen-yogurt cones revealed the distribution in the table. The owner wishes to know whether or not the pattern of sales for frozen-yogurt cones differs from that of ice-cream cones. For a goodness-of-fit test based on the χ² distribution, which of the following is the expected count for the cell corresponding to two scoops?

• 84

• 75

• 83.33

From experience, the owner of an ice-cream shop has found that 60% of all sales of ice-cream cones are for one-scoop cones, 30% are for two-scoop cones, and the remaining 10% are for three-scoop cones. Recently, the shop added frozen-yogurt cones to its menu. A random sample of 250 sales of frozen-yogurt cones revealed the distribution in the table. The owner wishes to know whether or not the pattern of sales for frozen-yogurt cones differs from that of ice-cream cones. For a goodness-of-fit test based on the χ² distribution, which of the following is the value of the χ² statistic?

• 0.2855

• 7.947

• 15.152

A fast-talking salesman offers you a “random digits” smartphone app that comes with a “guarantee” of randomness. You don’t trust him, so you insist on testing the app by generating a sample of 100 digits. Here are the resulting frequencies of the ten digits 0 through 9 in your sample of size 100. With the salesman getting more nervous by the minute, you then conduct a goodness-of-fit test of the null hypothesis that the app’s digits are indeed random (that is, that the probability of seeing any of the ten digits 0 through 9 in a single observation is equal to 1/10 or 0.10) against the alternative that they are not random. Which one of the following analyses of the conditions for performing this test is correct?

• Since the data do not come from a simple random sample of possible digits produced by the app, the conditions for performing the test have not been met.

• Since all the observed counts are not greater than 5, the conditions for performing the test have not been met.

• All conditions for performing this test have been satisfied.

A fast-talking salesman offers you a “random digits” smartphone app that comes with a “guarantee” of randomness. You don’t trust him, so you insist on testing the app by generating a sample of 100 digits. These are the resulting frequencies of the ten digits 0 through 9 in your sample of size 100. With the salesman getting more nervous by the minute, you then conduct a goodness-of-fit test of the null hypotheses that the app's digits are indeed random (that is, that the probability of seeing any of the ten digits 0 through 9 in a single observation is equal to 1/10 or 0.10) against the alternative that they are not random. The χ² statistic for this test is 11.2. Which of the following conclusions regarding the null hypothesis is appropriate?

• Reject H₀ at either the α = 0.05 or α = 0.01.

• Reject H₀ at the α = 0.05; fail to reject H₀ at the α = 0.01.

• Fail to reject H₀ at either the α = 0.05 or α = 0.01.

A study was done to examine the personal goals of elementary school children. A random sample of 500 students was selected from schools in the state of Georgia. The students received a questionnaire regarding personal goals. They were asked what they would most like to do at school: get good grades, be popular, or be good at sports. The results are presented in the table, classified by the gender of the child. We wish to test the null hypothesis that gender and personal goal are independent. Under the null hypothesis, the expected number of boys that would select “be good at sports” is

• 59.5

• 67

• 74

A study was done to examine the personal goals of elementary school children. A random sample of 500 students was selected from schools in the state of Georgia. The students received a questionnaire regarding personal goals. They were asked what they would most like to do at school: get good grades, be popular, or be good at sports. The results are presented in the table, classified by the gender of the child. We wish to test the null hypothesis that gender and personal goal are independent. Which one of the following is the correct χ² component for the "Boys/Be good at sports" cell?

• 20

• 34.5

• 50.88

In a study of the responsiveness of small businesses in different locations, questionnaires were sent to 200 randomly chosen small businesses in rural and small-town locations, 200 randomly chosen small businesses in suburban areas, and 200 randomly chosen small businesses in urban areas. The numbers of businesses of each type that responded to the questionnaire were recorded. Here is a table of the results. We wish to know whether the data provide sufficient evidence to conclude that the distributions of response rates for small businesses in these three locations are not all the same. The value of the χ² test for homogeneity is 12.988. The P-value of this test is

• between 0.001 and 0.0025.

• between 0.01 and 0.025.

• between 0.025 and 0.05.

A study was conducted to determine whether or not there was a relationship between a child’s birth order and his or her chances of exhibiting delinquent behavior. The subjects were a random sample of girls enrolled in public high schools in a large city. Each subject filled out a questionnaire that measured whether or not she had shown evidence of delinquent behavior, as well as her birth order. The resulting data are given in the table. Which one of the following is the appropriate null hypothesis for this situation?

• The distribution of delinquent behavior is the same for each birth order.

• Delinquent behavior and birth order are independent.

• The distribution of birth order is the same for those subjects exhibiting delinquent and those who do not.