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Chapter 12: Q. 4 (page 520)

Explain why a chi-square goodness-of-fit test, a chi-square independence test, or a chi-square homogeneity test is always right tailed.

Short Answer

Expert verified

For all three tests, the null hypothesis is rejected only when the observed and expected frequencies match up poorly, which corresponds to large values of the chi-square test statistics, resulting in all the three tests are always right tailed.

Step by step solution

01

Step 1. Explain main properties of the chi-square distributions.

The properties of the chi-square distributions is given below,

(i) The research hypothesis for the chi-square is always a one-tailed test. Hence, the distributions are positively skewed.

(ii) Chi-square values are always positive. The minimum possible value is zero, and there is no upper limit to its maximum value. A chi-square of zero means that the variables are completely independent and the observed frequencies in every cell are equal to the corresponding expected frequencies.

(iii) As the number of degrees of freedom increases, the chi-square distribution becomes more symmetrical and with degrees of freedom greater than 30, begins to resemble the normal curve.

02

Step 2. Explain the reason.

Considering all the above given properties,

When the observed and expected frequencies match up poorly then only the null hypothesis is rejected, corresponding to larger values of the chi-square test statistic.

Therefore, all the three tests are always right tailed.

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Most popular questions from this chapter

US. Hospitals. Refer to Exercise 12.50.


24 or fewer25-7475 or moreTotal
General260158635575403
Psychiatric24242471737
Chronic132226
Tuberculosis0224
Other25177208410
Total310201042606580

a. Determine the conditional distribution of number of beds within each facility type.

b. Does an association exist between facility type and number of beds for U.S. hospitals? Explain your answer.

c. Determine the marginal distribution of number of beds for U.S. hospitals.

d. Construct a segmented bar graph for the conditional distributions and marginal distribution of number of beds. Interpret the graph in light of your answer to part (b),

e. Without doing any further calculations, respond true or false to the following statement and explain your answer. "The conditional distributions of facility type within number-of-beds categories are identical.

f. Determine the marginal distribution of facility type and the conditional distributions of facility type within number-of-beds categories.

g. What percentage of hospitals are general facilities?

h. What percentage of hospitals that have at least 75 beds are general facilities?

i. What percentage of general facilities have at least 75 beds?

In each of Exercises 12.18-12.23, we have provided a distribution and the observed frequencies of the values of a variable from a simple random sample of a population. In each case, use the chi-square goodness-of-fit test to decide, at the specified significance level, whether the distribution of the variable differs from the given distribution.

Distribution: 0.2, 0.4, 0.3, 0.1

Observed frequencies: 85, 215, 130, 70

Significance level = 0.05

The Quinnipiac University Pol conducts nationwide surveys as a public service and for research. This problem is baed on the results of one such poll that asked independent random samples of American adults in urban, suburban, and rural regions, "Do you support or oppose requiring background checks for all gun buyers?" Here are the results.

At the 1%significance level, do the data provide sufficient evidence to conclude that a difference exists in the proportions of supporters among the three regions?

In each of the given Exercises, we have given the number of possible values for two variables of a population. For each exercise, determine the maximum number of expected frequencies that can be less than 5 in order that Assumption 2 of Procedure 12.2 on page 506 to be satisfied. Note: The number of cells for a contingency table with m rows and n columns is m鈰卬.

12.71 two and three

In each of Exercises 12.11-12.16, we have given the relative frequencies for the null hypothesis of a chi-square goodness-of-fir text and the sample size. In each case, decide whether Assumptions 1 and 2 for using that text are satisfied.

Sample size : n= 100.

Relative frequencies: 0.44 , 0.25 , 0.30 , 0.01.
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