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2/3 Examination: Answers
Answers to Sample Questions for 2/3 Exam from Previous Years
 
 
Which test statistic --- (1) z-score, (2) t-score, (3) F ratio ---- would be most suitable for testing relationships between
these pairs of variables?
 
1.___2___ Difference in mens' and womens' attendance rates at religious services
(The t-test, which produces a t-score, is for difference between means of two nominal groups. No Ns are mentioned so t-test is appropriate.)

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2.___1___ Difference in the predicted mean Republican vote in 1998 by the Gallup poll with 900 interviews and the CBS poll with 1,200.
(One could also use a t-test, but the samplesizes are so large that the test statistic would distribute normally, so a z-score is appropriate.)

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3.___3___ Literacy rates by countries and a classification of countries by region.
(Literacy rates are measured on a continuous scale, and region is a nominal variable. So ANOVA is appropriate, and it produces an F ratio.)

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4. The formula provides an unbiased estimate of the population:
1- standard deviation
(Dividing by N-1 corrects for degrees of freedom.)

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5. The standard error of the mean is
3- the standard deviation of the sampling distribution of means
(Remember that a standard error of any statistic is, by definition, the standard deviation of a distribution of an infinite number of samples drawn to compute that statistic.)

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6. The standard error of the mean is equal to
(In this expression, the numerator and denominator have been squared.)

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7. Given a sample size of 100, mean of 36, population standard deviation of 5, and confidence level of .95, the confidence interval for the mean of the population is closest to
4- 35 to 37
(This requires applying the formula for the standard error of the mean: . So the standard error is .5. You don't simply add and subtract .5 from your sample mean of 36, for that would mark off only one standard error above and below the mean. The rule of thumb for the 95% confidence interval is to double the standard error and then add and subtract it from your estimate.)

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8. In practical terms, the amount of accuracy in predicting from national samples to the U.S. Population is least dependent on
1- the proportion that the sample is of the population
2- The "absolute number of cases in the sample" is N, and is critical to sample accuracy.
3- "The population variance for the attribute being estimated," is used in the numerator of the formula for the standard error.
4- "The procedure used in drawing the sample," is essential, as each case must have a known probability of being selected. The % the sample is of the population gets to be important only when the sample grows to about 20% of the population, which rarely occurs in social science research.)
9. Which of the following terms does not belong with the other terms?
4- z-score
("Type I error," "alpha level," and "region of rejection" all pertain to the risk of rejecting a true null hypothesis.)

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10. Which of the following statistical distributions is interpreted with reference to two different degrees of freedom?
1- F
(Tables of the t distribution are entered using only one value for degrees of freedom. Degrees of freedom are irrelevant to a "skewed" distribution. A normal distribution is not dependent on degrees of freedom.)

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11. If one has a directional hypothesis, he or she especially wants to
1- make a one-tailed test
(A one tailed test allows for rejecting the null with a smaller critical value, which means observing a smaller difference than required with a two tailed test.)

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12. The sampling distribution of the difference between two means for independent samples --
4- tends to become normal as the sample sizes increase.
(This is the gist of the Central Limit Theorem.)

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13. In the analysis of variance model, the null hypothesis of no difference among the means for the subgroups will tend to be rejected as
1- the "between group" variance exceeds the "within group" variance.
(The null hypothesis asserts that the groups could be drawn from the same parent population. This implies that they have similar means and similar variances. To the extent that differences "between" groups increases, the groups groups tend to be different from each other.)

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14. Which statement is true about data expressed in proportions?
3- Proportions computed for a dichotomized variable are simply a type of mean.
(Suppose you coded a dichotomy, such as sex, Male=0 and Female=1. Suppose further that you have a group of 10 people, 4 of whom are males. The mean for "sex" would be .6, and the proportion would be .6.)

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15. Another name for "between groups" sum of squares in the analysis of variance is
3- explained variation
(Between groups sum of squares is calculated on the difference between the group means on a dependent variable and the grand mean. Big differences indicate that the groupings of cases has a strong effect on the values of the dependent variable.)

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16. An unbiased estimator of a population parameter is
1- a statistic whose mean value of its own sampling distribution is equal to the population parameter.
(Thus, the mean (average) value of the sampling distribution of the statistic being estimated is equal to the mean (average) value in the population.)

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17. Calculate the variance for this set of numbers: -5,-5,-5,5,5,5
Case
X
mean of X
(x - mean)
(x - mean)2
1
-5
0
-5
25
2
-5
0
-5
25
3
-5
0
-5
25
4
5
0
5
25
5
5
0
5
25
6
5
0
5
25

 

(x-mean)2=

150
Variance = 150 / 6 =__25___

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18. According to the table of areas under the normal curve, what positive and negative z-score value would be used to establish the region of rejection for a TWO-tailed test nearest to the .04 level of significance?
z-score = ___2.05______
(A z-score of 2.05 marks 0.0202 in one tail. So a two-tailed test would encompass 0.0404 in both tails.)

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Here is T-TEST output comparing Democrat and Republican Members of Congress, with the dependent variable being a scale of increasing "conservatism" in voting on social issues. Answer questions 19 and 20.
 VARIABLE         NUMBER              STANDARD    STANDARD
                 OF CASES     MEAN   DEVIATION      ERROR 
    DEMOCRATS      248      30.5806     23.441      1.488 
    REPUBLICANS    180      65.7278     16.318      1.216 
                     
                Mean Difference = -35.1472
                     
                Levene's Test for Equality of Variances:F= 2.06  P= .000
   
   Variances    t-value     df    2-TailSig     
   
    Equal        -17.30     426      .000  
    Unequal      -18.28     425.31   .000
 
19.Which expression below was used to compute the degrees of freedom for Equal Variance estimate of the T-Test above?
4- (N1 + N2 - 2)
(Check it out. df = 248 + 180 - 2 = 426)

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20. Which statement about the above T-TEST output is true? Return
3- The F-value is significant at the .05 level, indicating that the population variance are equal and that you are advised to use the Unequal variance estimate rather than the Equal estimate in checking the significance of t.
 
 
Below is SPSSx output from MEANS for GNP PER CAPITA in the POLITY data analyzed by type of government. Answer questions 21, 22, 23, and 24 with reference to this output.
 
   DEPENDENT VARIABLE   GNPPCA80   GNP PER CAPITA IN US DOLLARS-1980
 INDEPENDENT VARIABLE   POLTYPE    TYPE OF POLITICAL SYSTEM
  
 VARIABLE      VALUE  LABEL                      MEAN    STD DEV    CASES 
  
 FOR ENTIRE POPULATION                      3633.0864  4267.0914       81   
 POLTYPE        1.00  PARLIAMENT/DEMOCRACY  6511.4815  4608.2098       27 
 POLTYPE        2.00  DEMOCRATIC & FEDERAL  9598.0000  5869.8825        5 
 POLTYPE        3.00  MULTIPARTY DEMOCRACY  1322.8571  1334.9746        7 
 POLTYPE        4.00  DEMOCRATIC FORM       2210.0000   113.1371        2 
 POLTYPE        5.00  MILITARY GOVERNMENT    895.8333   900.6408       12 
 POLTYPE        8.00  ONEPARTY COMMUNIST    3587.5000  2431.3180        8 
 POLTYPE        9.00  ONEPARTY SOCIALIST    2270.0000   494.9747        2  
 POLTYPE       10.00  ONEPARTY MARXIST       276.6667   140.1190        3 
 POLTYPE       11.00  ONEPARTY STATE         798.6667  1015.8168       15 
                        
                        SUM OF                MEAN
 SOURCE                 SQUARES    D.F.      SQUARE       F     SIG.
  
 BETWEEN GROUPS     690959683.13     8  86369960.391   8.1217  .0000
 WITHIN GROUPS      765685845.26    72  10634525.629  
 
21. Showing the math, compute the MEAN SQUARE for BETWEEN GROUPS to one decimal: _____________
Between groups mean square = 690959683.13 / 8 = 86369960.391

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22. Showing the math, compute the F ratio to one decimal: _____________
 
F = 86369960.391 / 10634525.629 = 8.1217

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23. Which statement gives the most accurate assessment of that relationship?
3- The null hypothesis can be rejected at the .01 level.

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24. Under the null hypothesis, the expected value of Pearson's r is
4- zero
(That is, the null hypothesis asserts that the two variables are not correlated. So any correlation you observe, may have observed by chance.)

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25. What condition exists when the variance of Y remains the same for different values of X?
1- homoscedasticity

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26. This formula for testing the significance of r: , assumes that

4- the test is against the null hypothesis that the population correlation, r (rho), is equal to 0. Return

 
27. If data for 2 variables are transformed into z-scores and then correlated, [This topic was not covered.]
1- the product moment correlation will be equal to the beta coefficient in the associated regression equation.
2- the beta coefficient will be equal to the b-coefficient in the regression equation.
3- the intercept in the regression equation will be equal to 0.
4- all of the above are true.

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28.Tested against the null hypothesis, a highly significant correlation merely indicates that
1- it is unlikely that there is no relationship between the variables in the universe.

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29. Which statistic is most appropriate for testing the significance of the difference among means for four independent samples?
3- F ratio

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30. The condition of heteroscedasticity is best shown in which diagram?
1 This one
(By convention, the base line represents the X-variable and the perpendicular line the Y-variable. Q. 25 above defines "homoscedasticity." Heteroscedasticity obtains when the variance of Y is not constant for different values of X.)

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