Deck 15: Chi-Squared Tests Optional

Question 1

Suppose that a random sample of 150 observations was drawn from a population. After calculating the mean and standard deviation, each observation was standardized and the number of observations in each of the intervals below was counted. Can we infer at the 5% significance level that the data were drawn from a normal population?
$$\begin{array} { | l | c| } \hline \text { Intervals } & \text { Frequency } \\\hline Z \leq - 1.5 & 15 \\- 1.5 < Z \leq - .5 & 32 \\- .5 \leq Z \leq .5 & 65 \\.5 < Z \leq 1.5 & 25 \\Z > 1.5 & 13 \\\hline\end{array}$$

Answer 1

H₀: p₁ = 0.0668, p₂ = 0.2417, p₃ = 0.3830, p₄ = 0.2417, p₅ = 0.0668 (The population is normal)
H₁: At least two proportions differ from their specified values (The population is not normal)
Rejection region: $\chi$ ² > $\chi$ ²_0.05,2 = 5.991
Test statistic: $\chi$ ² = 8.347
Conclusion: Reject the null hypothesis. We can't infer at the 5% significance level that the data were drawn from a normal population.

Question 2

We can use the goodness-of-fit test to determine whether data were drawn from any distribution of interest. The most common application of this procedure is a test of ____________________.

Answer 2

normality

Question 3

To test for normality, the ____________________ hypothesis specifies probabilities of certain intervals within the normal distribution.

Answer 3

null
H₀

Question 4

To test for normality, the ____________________ hypothesis is that at least two proportions differ from their specified values.

Question 5

A large value of the chi-squared test statistic in a test of normality means you reject H₀ and conclude that the data ____________________ (do/do not) come from a normal distribution.

Question 6

The number of degrees of freedom associated with the chi-squared test statistic for normality is the number of ____________________ minus 1 minus the number of ____________________ estimated.

Question 7

In a goodness-of-fit test, the null hypothesis states that the data came from a normally distributed population. The researcher estimated the population mean and population standard deviation from a sample of 200 observations. In addition, the researcher used 5 standardized intervals to test for normality. Using a 10% level of significance, the critical value for this test is 4.60517.

Question 8

The chi-squared test for normality must follow the rule of ____________________ regarding expected values.

Question 9

The number of degrees of freedom in a chi-squared test for normality, where the number of standardized intervals is 5 and there are 2 population parameters to be estimated from the data, is equal to:

A)5
B)4
C)3
D)2

Question 10

The null hypothesis states that the sample data came from a normally distributed population. The researcher calculates the sample mean and the sample standard deviation from the data. The data arrangement consisted of five categories. Using $\alpha$ = 0.05, the appropriate critical value for this chi-squared test for normality is 5.99147.

Question 11

If we want to perform a one-tail test of a population proportion p, we can employ either the z-test of p, or the chi-squared goodness-of-fit test.

Question 12

The number of degrees of freedom in testing for normality is the:

A)number of intervals used to test the hypothesis minus one.
B)number of parameters estimated minus one.
C)number of intervals used to test the hypothesis minus one minus the number of parameters estimated.
D)None of these choices.

Question 13

Suppose that a random sample of 60 observations was drawn from a population. After calculating the mean and standard deviation, each observation was standardized and the number of observations in each of the intervals below was counted. Can we infer at the 10% significance level that the data were drawn from a normal population?
$$\begin{array} { | l | c | } \hline \text { Intervals } & \text { Frequency } \\\hline Z \leq - 1 & \mathbf { 8 } \\- 1 < Z \leq 0 & 30 \\0 < Z \leq 1 & 17 \\Z > 1 & 5 \\\hline\end{array}$$

Question 14

If we want to test for differences between two populations of nominal data with exactly two categories, we can employ either the z-test of p₁-p₂, or the chi-squared test of a contingency table.

Question 15

If we want to perform a two-tail test of a population proportion p, we can only use the z-test of p.

Question 16

In a goodness-of-fit test, the null hypothesis states that the data came from a normally distributed population. The researcher estimated the population mean and population standard deviation from a sample of 500 observations. In addition, the researcher used 6 standardized intervals to test for normality. Using a 5% level of significance, the critical value for this test is:

A)11.1433
B)9.3484
C)7.8147
D)9.4877

Question 17

In a goodness-of-fit test, the null hypothesis states that the data came from a normally distributed population. The researcher estimated the population mean and population standard deviation from a sample of 300 observations. In addition, the researcher used 6 standardized intervals to test for normality. Using a 2.5% level of significance, the critical value for this test is 14.4494.

Question 18

When the problem objective is to describe a population of nominal data with exactly two categories, we can employ either the z-test of a population proportion p, or the chi-squared goodness-of-fit test.

Question 19

The following data are believed to have come from a normal probability distribution. $$\begin{array} { l l l l l l l l l } 26 & 21 & 25 & 20 & 21 & 29 & 26 & 23 & 22 \\24 & 30 & 23 & 32 & 26 & 24 & 32 & 16 & 36 \\21 & 31 & 26 & 23 & 32 & 35 & 40 & 30 & 14 \\46 & 27 & 33 & 25 & 27 & 21 & 26 & 18 & 29\end{array}$$ The mean of this sample equals 26.80, and the standard deviation equals 6.378. Use the goodness-of-fit test at the 5% significance level to test whether the data indeed come from a normal distribution.

Question 20

The number of degrees of freedom associated with the chi-squared test for normality is the number of intervals used minus the number of parameters estimated from the data.

Question 21

If we want to conduct a two-tail test of a population proportion, we can employ:

A)z-test of a population proportion.
B)the chi-squared test of a binomial experiment since z² = $\chi$ ².
C)the chi-squared test of a contingency table.
D)Both a and b

Question 22

If we want to conduct a one-tail test of a population proportion, we can employ:

A)z-test of a population proportion.
B)the chi-squared test of a binomial experiment since z² = $\chi$ ².
C)the chi-squared test of a contingency table.
D)Both a and b

Question 23

The squared difference between the observed and expected frequencies should be large if there is a significant difference between the proportions.

Question 24

Which of the following tests is used to analyze nominal data?

A)The z-test for one proportion, p, or difference of two proportions, p₁ - p₂.
B)The chi-squared goodness-of-fitness test.
C)The chi-squared test of a contingency table.
D)All of these choices are true.

Question 25

When we test for differences between two populations of nominal data with two categories, we can use only one technique, namely, the chi-squared test of a contingency table.

Question 26

Which of the following tests is appropriate for nominal data if the problem objective is to compare two populations and the number of categories exceeds 2?

A)The z-test for one proportion, p, or difference of two proportions, p₁ - p₂.
B)The chi-squared goodness-of-fit test.
C)The chi-squared test of a contingency table.
D)All of these choices are true.

Question 27

If we want to perform a one-tail test for differences between two populations of nominal data with exactly two categories, we must employ the z-test of p₁ -p₂.

Question 28

Which of the following statements is true for chi-squared tests?

A)Testing for equal proportions is identical to testing for goodness-of-fit.
B)The number of degrees of freedom in a test of a contingency table with r rows and c columns is (r - 1)(c - 1).
C)The number of degrees of freedom in a goodness-of-fit test with k categories is k - 1.
D)All of these choices are true.

Question 29

Which of the following tests is appropriate for nominal data if the problem objective is to compare two or more populations and the number of categories is at least 2?

A)The z-test for one proportion, p, or difference of two proportions, p₁ - p₂.
B)The chi-squared goodness-of-fitness test.
C)The chi-squared test of a contingency table.
D)All of these choices are true.

Question 30

If we want to perform a two-tail test for differences between two populations of nominal data with exactly two categories, we can employ either the z-test of p₁ - p₂, or the chi-squared test of a contingency table.

Question 31

When we describe a population of nominal data, with exactly two categories, the multinomial experiment is actually a binomial experiment with one of the categorical outcomes labeled "success" and the other labeled "failure".

Question 32

Suppose that two shipping companies, A and B, each decide to estimate the annual percentage of shipments on which a $100 or greater claim for loss or damage was filed by sampling their records, and they report the data shown below. $$\begin{array} { | l | c c | } \hline & \text { Company A } & \text { Company B } \\\hline \text { Total shipments sampled } & \text { 800 } & \text { 600 } \\\text { Number of shipments with a claim } \geq \$ 100 & 200 & 100 \\\hline\end{array}$$ The owner of Company B is hoping to use these data to show that her company is superior to Company A with regard to the percentage of claims filed. Which test would be used to properly analyze the data in this experiment?

A)The $\chi$ ² test of a contingency table in a two-way contingency table.
B)The $\chi$ ² goodness-of-fitness test.
C)The z-test for the difference in two proportions.
D)The ANOVA F test for main treatment effect.

Question 33

If we use the chi-squared method of analysis we must first check that there are at least 5 observations in each cell of the contingency table.

Question 34

To describe a population with more than two categories you can only use a chi-squared goodness-of-fit test.

Question 35

Mathematical statisticians have established that if we square the value of z, the test statistic for the test of one proportion p, we produce the $\chi$ ² statistic. That is, z² = $\chi$ ².

Question 36

A test for the differences between two proportions can be performed using the chi-squared distribution.

Question 37

The chi-squared distribution is used in:

A)a goodness-of-fit test.
B)a test of a contingency table.
C)describing a population having more than two categories.
D)All of these choices are true.

Question 38

Which of the following tests does not use the chi-squared distribution?

A)Test of a contingency table.
B)Goodness-of-fit test.
C)One tailed test for two proportions.
D)None of these choices.

Question 39

In testing the difference between two proportions using the normal distribution, we may use either a one-tailed or two-tailed test.

Question 40

A test for whether one proportion is higher than the other can be performed using the chi-squared distribution.

Question 41

To analyze the relationship between two nominal variables, which test(s) can you use?

Question 42

If you want to compare two populations that each have two categories, you can use a z-test for two proportions, or a chi-squared test of a(n) ____________________.

Question 43

If you want to describe a population with more than two categories, which test(s) can you use?

Question 44

For comparing two or more populations each having two or more categories, you can use which test(s)?

Question 45

There are two critical factors in identifying the technique used when the data are nominal. The first is the problem objective. The second is the number of ____________________ that nominal variable can assume.

Question 46

If you want to describe a population with two categories, you can use a(n) ____________________ test of p or the chi-squared goodness-of-fit test.

Question 47

If you want to compare two populations that each have two categories, which test(s) can you use?

Question 48

If you want to compare two populations that each have more than two categories, which test(s) can you use?

Question 49

To produce expected values for a test of a contingency table, you multiply estimated joint probabilities for each cell by the total sample size, n.

Question 50

If you want to compare two populations that each have more than two categories, you can use a chi-squared test of a(n) ____________________.

Question 51

In a test of a contingency table, rejecting the null hypothesis concludes the variables are not independent.

Question 52

Which test(s) can you use when you want to describe a population with two categories?

Question 53

To calculate the expected values in a test of a contingency table, you assume that the null hypothesis is true.

Question 54

What statistic do we get when we square the value of z?

Question 55

If you want to describe a population with more than two categories, you can use a chi-squared ____________________ test.

Question 56

Which of the following tests is appropriate for nominal data if the problem objective is to describe a population with more than two categories?

A)z-test for one proportion, p, or difference of two proportions, p₁ - p₂.
B)chi-squared goodness-of-fitness test.
C)chi-squared test of a contingency table.
D)All of these choices are true.

Question 57

What are the two critical factors in identifying the technique used when the data are nominal?

Question 58

If we square the value of z (the test statistic in the test of a proportion) we produce the ____________________ statistic.

Question 59

For comparing two or more populations each having two or more categories, use a(n) ____________________ test of a(n) ____________________.

Question 60

To analyze the relationship between two nominal variables, use a(n) ____________________ test of a(n) ____________________.

Question 61

A chi-squared test of a contingency table is applied to a contingency table with 3 rows and 4 columns for two qualitative variables. The degrees of freedom for this test must be 12.

Question 62

The test statistic for the chi-squared test of a contingency table is the same as the test statistic for the goodness-of-fit test.

Question 63

A chi-squared test of a contingency table with 10 degrees of freedom results in a test statistic of 17.894. Using the chi-squared table, the most accurate statement that can be made about the p-value for this test is that 0.05 < p-value < 0.10.

Question 64

Contingency tables are used in:

A)testing independence of two samples.
B)testing dependence in matched pairs.
C)testing independence of two qualitative variables in a population.
D)describing a single population.

Question 65

How do you find the probabilities needed to obtain expected frequencies for a test of a contingency table?

A)If there are r rows and c columns, let each probability be 1/rc.
B)Use the probabilities specified in the null hypothesis.
C)Assume H₀ is true and use your data to calculate them.
D)None of these choices.

Question 66

The degrees of freedom for the test statistic in a test of a contingency table is (r - 1)(c -1) where r is the number of rows in the table, and c is the number of columns.

Question 67

To address whether two variables are related in a contingency table, the alternative hypothesis, H₁, is:

A)The two variables are independent.
B)The two variables are dependent.
C)The two variables are equal.
D)None of these choices.

Question 68

In the test of a contingency table, the observed cell frequencies must satisfy the rule of 5.

Question 69

If you reject H₀ in a test of a contingency table, you conclude that based on your data:

A)The two nominal variables are independent.
B)The two nominal variables are equal.
C)The two nominal variables have the same proportions listed in H₀.
D)None of these choices.

Question 70

A chi-squared test of a contingency table is applied to a contingency table with 4 rows and 4 columns for two qualitative variables. The degrees of freedom for this test must be 9.

Question 71

If two events A and B are independent, the P(A and B) = P(A) + P(B).

Question 72

The expected frequency for the cell in row i and column j is the row i total plus the row j total, all divided by n.

Question 73

In a goodness-of-fit test, H₀ lists specific values for proportions and the test of a contingency table does not.

Question 74

A chi-squared test of a contingency table with 6 degrees of freedom results in a test statistic of 13.25. Using the chi-squared table, the most accurate statement that can be made about the p-value for this test is that p-value is greater than 0.025 but smaller than 0.05.

Question 75

A large chi-squared test statistic in a test of a contingency table means you conclude:

A)The two nominal variables are dependent.
B)The two nominal variables are equal.
C)The two nominal variables have the same proportions listed in H₀.
D)None of these choices.

Question 76

In a chi-squared test of a contingency table, the value of the test statistic was $\chi$ ² = 15.652, and the critical value at $\alpha$ = 0.025 was 11.1433. Thus, we must reject the null hypothesis at $\alpha$ = 0.025.

Question 77

In the test of a contingency table, the expected cell frequencies must satisfy the rule of 5.

Question 78

To address whether two variables are related in a contingency table, the null hypothesis, H₀, says that

A)The two variables are independent.
B)The two variables are dependent.
C)The two variables are equal.
D)None of these choices.

Question 79

The number of degrees of freedom for a contingency table with r rows and c columns is:

A)r + c
B)rc
C)(r - 1)(c -1)
D)None of these choices.

Question 80

The chi-squared test of a contingency table is based upon:

A)one quantitative variable.
B)two quantitative variables.
C)one qualitative variable.
D)two qualitative variables.