Question 1

Suppose that we wish to test the null hypothesis that for 3 cells, A, B, and C, the cell categories are equal. We observed 8 data in cell A, 13 in cell B, and 9 in cell C. What is the decision rule using the 0.05 significance level?

Accepted Answer

To test the null hypothesis, we can use the chi-square test of independence. With 3 cells, we have 2 degrees of freedom. Using a significance level of 0.05, the critical value of chi-square with 2 degrees of freedom is 5.991. If the calculated chi-square value is greater than 5.991, we reject the null hypothesis in favor of the alternative hypothesis that at least one cell category is different from the others. The calculated chi-square value can be found using a chi-square calculator or software.

Question 2

A distributor of personal computers has five locations in the city. The sales in units for the first quarter of the year were as follows:  What is the critical value at the 0.01 level of risk?

Accepted Answer

To find the critical value at the 0.01 level of risk, we need to use a t-distribution table with 4 degrees of freedom since we have 5 locations and are calculating the sample variance. From the table, the critical value is 2.776. However, since we are looking for a two-tailed test, we need to multiply this value by 2, giving us a final critical value of 5.552. Therefore, the closest answer is C, which is 13.277.

Question 3

To find the expected frequency in a contingency table:

Accepted Answer

To find the expected frequency in a contingency table, you need to multiply the row and column totals and divide by the grand total. This gives the expected number of observations in each cell assuming that there is no association between the variables.

Question 4

Which of the following are correct statements regarding the goodness-of-fit test?

Accepted Answer

The goodness-of-fit test can be used with data of nominal scale. However, options B and C are incorrect, as the population does not need to be normal and not all expected frequencies need to be equal for the goodness-of-fit test to be used.

Question 5

i. The chi-square distribution is positively skewed. ii. Nonparametric tests of hypothesis, which are also called distribution free tests, require the population to be normally distributed.
iii. The computed value of chi-square is always positive because the difference between the observed frequencies and the expected frequencies are squared.

Accepted Answer

- The chi-square distribution is indeed positively skewed.- Nonparametric tests do not require the population to be normally distributed; they are used when this assumption cannot be met.- The computed value of chi-square is always positive due to the squaring of differences between observed and expected frequencies.

Question 6

i. There is not one, but a family of chi-square distributions. There is a chi-square distribution for 1 degree of freedom, another for 2 degrees of freedom, another for 3 degrees of freedom, and so on. ii. The shape of the chi-square distribution depends on the size of the sample.
iii. Small differences between observed and expected frequencies are due to chance.

Accepted Answer

The chi-square distribution indeed varies with degrees of freedom, not the sample size, and small differences between observed and expected frequencies can indeed be attributed to chance. The shape of the chi-square distribution is determined by the degrees of freedom, not directly by the sample size.

Question 7

i. The shape of the chi-square distribution depends on the size of the sample. ii. Small differences between observed and expected frequencies are due to chance.
iii. The chi-square distribution with large degrees of freedom approaches a normal distribution.

Accepted Answer

The shape of the chi-square distribution depends on the degrees of freedom, not the sample size. Small differences can be due to chance, and with large degrees of freedom, the chi-square distribution approaches a normal distribution.

Question 8

i. Nonparametric tests of hypotheses, which are also called distribution free tests, require the population to be normally distributed. ii. The computed value of chi-square is always positive because the difference between the observed frequencies and the expected frequencies are squared.
iii. The shape of the chi-square distribution changes for each number of degrees of freedom.

Accepted Answer

(i) is false as nonparametric tests do not assume normal distribution.
(ii) is true as the squared differences between observed and expected frequencies are always positive.
(iii) is true as the shape of the chi-square distribution changes with the degrees of freedom.

Question 9

i. Nonparametric tests require no assumptions about the shape of the population distribution. ii. There is not one, but a family of chi-square distributions. There is a chi-square distribution for 1 degree of freedom, another for 2 degrees of freedom, another for 3 degrees of freedom, and so on.
iii. The chi-square distribution with large degrees of freedom approaches a normal distribution.

Accepted Answer

All three statements are correct. Nonparametric tests indeed do not assume a specific population distribution shape, there is a family of chi-square distributions each corresponding to different degrees of freedom, and the chi-square distribution does approach a normal distribution as the degrees of freedom increase.

Question 10

i. Tests of hypotheses for nominal or ordinal levels of measurement are called nonparametric or distribution-free tests. ii. There is not one, but a family of chi-square distributions. There is a chi-square distribution for 1 degree of freedom, another for 2 degrees of freedom, another for 3 degrees of freedom, and so on.
iii. The shape of the chi-square distribution depends on the size of the sample.

Accepted Answer

Statement (i) is true as nonparametric tests are used for nominal or ordinal levels of measurement.
Statement (ii) is also true as there are different chi-square distributions for different degrees of freedom.
However, statement (iii) is false as the shape of the chi-square distribution does not depend on the size of the sample, but on the degrees of freedom.

Question 11

i. The shape of the chi-square distribution changes for each number of degrees of freedom. ii. The minimum computed value of chi-square is zero.
iii. The chi-square distribution is a positively skewed distribution.

Accepted Answer

The shape of the chi-square distribution indeed changes with the number of degrees of freedom, the minimum value it can take is zero (since it's based on squared differences), and it is positively skewed, especially with lower degrees of freedom.

Question 12

i. The minimum computed value of chi-square is one. ii. The chi-square distribution is a positively skewed distribution.
iii. The lowest level of data for which the chi-square goodness-of-fit test is appropriate is the nominal level.

Accepted Answer

The minimum value of chi-square can actually be zero, not one, if observed frequencies perfectly match expected frequencies. The chi-square distribution is indeed positively skewed, especially with a low degree of freedom, and it becomes less skewed as the degrees of freedom increase. The chi-square goodness-of-fit test is indeed appropriate for nominal level data, as it compares observed frequencies to expected frequencies in categorical data.

Question 13

i. Nonparametric tests require no assumptions about the shape of the population distribution. ii. Tests of hypotheses for nominal or ordinal levels of measurement are called nonparametric or distribution-free tests.
iii. There is not one, but a family of chi-square distributions. There is a chi-square distribution for 1 degree of freedom, another for 2 degrees of freedom, another for 3 degrees of freedom, and so on.

Accepted Answer

(i) Nonparametric tests do not assume any specific shape of population distribution.
(ii) Nonparametric tests are suitable for nominal or ordinal levels of measurement as they are not based on assumptions about the distribution.
(iii) Chi-square distribution has a family of distributions for different degrees of freedom.

Question 14

i. Small differences between observed and expected frequencies are due to chance. ii. The chi-square distribution with large degrees of freedom approaches a normal distribution.
iii. The chi-square distribution is positively skewed.

Accepted Answer

All three statements are correct: (i) is a basic principle of chi-square tests, (ii) is a property of the chi-square distribution as degrees of freedom increase, and (iii) describes the shape of the chi-square distribution.

Question 15

What is our decision for a goodness-of-fit test with a computed value of chi-square of 1.273 and a critical value of 13.388?

Accepted Answer

Since the computed value of chi-square (1.273) is less than the critical value of 13.388, we cannot reject the null hypothesis.

Question 16

In a contingency table suppose that we are comparing males versus females against five glades: A, B, C, D and F. The degrees of freedom will be:

Accepted Answer

The answer of In a contingency table suppose that we...

Question 17

i. The chi-square distribution with large degrees of freedom approaches a normal distribution. ii. The chi-square distribution is positively skewed.
iii. Nonparametric tests of hypotheses, which are also called distribution free tests, require the population to be normally distributed.

Accepted Answer

Statement (i) is correct because as the degrees of freedom increase, the chi-square distribution approaches a normal distribution.
Statement (ii) is also correct because the distribution is positively skewed as it cannot take negative values and has more values on the right-hand side of the distribution.
Statement (iii) is incorrect because nonparametric tests do not rely on the assumption of normality in the population. These tests do not require any assumptions about the population distribution, unlike parametric tests.

Question 18

i. Nonparametric tests of hypotheses, which are also called distribution free tests, require the population to be normally distributed. ii. The shape of the chi-square distribution changes for each number of degrees of freedom.
iii. The chi-square distribution is a positively skewed distribution.

Accepted Answer

Nonparametric tests do not require the population to be normally distributed; that's one of their main advantages. The shape of the chi-square distribution does indeed change with the degrees of freedom, and it is always positively skewed.

Question 19

Suppose that we wish to test the null hypothesis that for 3 cells, A, B, and C, the cell categories are equal. We observed 8 data in cell A, 13 in cell B, and 9 in cell C. What is the decision regarding the null hypothesis?

Accepted Answer

The chi-square test for independence is used to test the null hypothesis that the cell categories are equal. The chi-square statistic is calculated as:```χ² = Σ (O - E)² / E```where O is the observed frequency and E is the expected frequency.The expected frequency for each cell is calculated as:```E = (row total * column total) / grand total```The grand total is the sum of all the observed frequencies.In this case, the row totals are 8, 13, and 9, the column totals are 8, 13, and 9, and the grand total is 30.The expected frequencies are:```E(A) = (8 * 8) / 30 = 2.13E(B) = (8 * 13) / 30 = 3.47E(C) = (8 * 9) / 30 = 2.40```The chi-square statistic is:```χ² = [(8 - 2.13)² / 2.13] + [(13 - 3.47)² / 3.47] + [(9 - 2.40)² / 2.40] = 10.64```The degrees of freedom for the chi-square test is (r - 1) * (c - 1), where r is the number of rows and c is the number of columns. In this case, r = 3 and c = 3, so the degrees of freedom is 4.The critical value for the chi-square test with 4 degrees of freedom and a significance level of 0.05 is 9.488.Since the chi-square statistic (10.64) is greater than the critical value (9.488), we reject the null hypothesis and conclude that the cell categories are not equal.

Question 20

i. The computed value of chi-square is always positive because the difference between the observed frequencies and the expected frequencies are squared. ii. The shape of the chi-square distribution changes for each number of degrees of freedom.
iii. The minimum computed value of chi-square is zero.

Accepted Answer

All three statements are correct. The chi-square value is always positive due to squaring the differences between observed and expected frequencies, its distribution shape changes with degrees of freedom, and the minimum value is zero, indicating no difference between observed and expected frequencies.

Quiz 14: Chi-Square Applications

Suppose that we wish to test the null hypothesis that for 3 cells, A, B, and C, the cell categories are equal. We observed 8 data in cell A, 13 in cell B, and 9 in cell C. What is the decision rule using the 0.05 significance level?

A distributor of personal computers has five locations in the city. The sales in units for the first quarter of the year were as follows: What is the critical value at the 0.01 level of risk?

To find the expected frequency in a contingency table:

Which of the following are correct statements regarding the goodness-of-fit test?

i. The shape of the chi-square distribution depends on the size of the sample. ii. Small differences between observed and expected frequencies are due to chance. iii. The chi-square distribution with large degrees of freedom approaches a normal distribution.

i. The shape of the chi-square distribution changes for each number of degrees of freedom. ii. The minimum computed value of chi-square is zero. iii. The chi-square distribution is a positively skewed distribution.

i. The minimum computed value of chi-square is one. ii. The chi-square distribution is a positively skewed distribution. iii. The lowest level of data for which the chi-square goodness-of-fit test is appropriate is the nominal level.

i. Small differences between observed and expected frequencies are due to chance. ii. The chi-square distribution with large degrees of freedom approaches a normal distribution. iii. The chi-square distribution is positively skewed.

What is our decision for a goodness-of-fit test with a computed value of chi-square of 1.273 and a critical value of 13.388?

In a contingency table suppose that we are comparing males versus females against five glades: A, B, C, D and F. The degrees of freedom will be:

i. The chi-square distribution with large degrees of freedom approaches a normal distribution. ii. The chi-square distribution is positively skewed. iii. Nonparametric tests of hypotheses, which are also called distribution free tests, require the population to be normally distributed.

i. Nonparametric tests of hypotheses, which are also called distribution free tests, require the population to be normally distributed. ii. The shape of the chi-square distribution changes for each number of degrees of freedom. iii. The chi-square distribution is a positively skewed distribution.

Suppose that we wish to test the null hypothesis that for 3 cells, A, B, and C, the cell categories are equal. We observed 8 data in cell A, 13 in cell B, and 9 in cell C. What is the decision regarding the null hypothesis?

i. The computed value of chi-square is always positive because the difference between the observed frequencies and the expected frequencies are squared. ii. The shape of the chi-square distribution changes for each number of degrees of freedom. iii. The minimum computed value of chi-square is zero.