Quantitative Methods II

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Quantitative Methods II - Quiz

Quantitative Methods II


Questions and Answers
  • 1. 

    The expected value of the difference of two sample means equals the difference of the corresponding population means:

    • A.

      Only if the populations are normally distributed

    • B.

      Only if the samples are independent

    • C.

      Only if the populations are approximately normal and the sample sizes are large

    • D.

      All of the above

    Correct Answer
    D. All of the above
    Explanation
    The expected value of the difference of two sample means equals the difference of the corresponding population means only if the populations are normally distributed, only if the samples are independent, and only if the populations are approximately normal and the sample sizes are large. These conditions ensure that the sampling distribution of the difference in means follows a normal distribution, allowing for accurate estimation of the population mean difference. Therefore, all of the above conditions must be met for the expected value of the difference of two sample means to equal the difference of the corresponding population means.

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  • 2. 

    In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference  is normal if:

    • A.

      The sample sizes are both greater than 30

    • B.

      The populations are normal

    • C.

      The populations are non-normal and the sample sizes are large

    • D.

      All of the above

    Correct Answer
    B. The populations are normal
    Explanation
    The sampling distribution of the sample mean difference is normal if the populations are normal. This is because when the populations are normal, the distribution of sample means will also be normal. The Central Limit Theorem states that as sample size increases, the distribution of sample means approaches a normal distribution regardless of the shape of the population. Therefore, if the populations are normal, the sampling distribution of the sample mean difference will also be normal.

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  • 3. 

    In testing for differences between the means of two independent populations the null hypothesis is:

    • A.

      H0: μ1-μ2=2

    • B.

      H0: μ1-μ2=0

    • C.

      H0: μ1-μ2>0

    • D.

      H0: μ1-μ2

    Correct Answer
    B. H0: μ1-μ2=0
    Explanation
    The correct answer is H0: μ1-μ2=0. In hypothesis testing for the difference between the means of two independent populations, the null hypothesis assumes that there is no difference between the means (μ1-μ2=0). This means that any observed difference between the sample means is due to random sampling variability, rather than a true difference in the population means. The alternative hypothesis, on the other hand, would state that there is a significant difference between the means (e.g., H1: μ1-μ2≠0 or H1: μ1-μ2>0 or H1: μ1-μ2

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  • 4. 

    When testing H0: μ12=0 vs.H1: μ12<0 , the observed value of the z-score was found to be –2.15.  The p - value for this test would be:

    • A.

      .0158

    • B.

      .0316

    • C.

      .9842

    • D.

      .9684

    Correct Answer
    A. .0158
    Explanation
    When testing the hypothesis H0: μ1-μ2=0 vs. H1: μ1-μ2

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  • 5. 

    In testing for differences between the means of two dependent populations the null hypothesis is:

    • A.

      H0: μD=2

    • B.

      H0: μD=0

    • C.

      H0: μD

    • D.

      H0: μD>0

    Correct Answer
    B. H0: μD=0
    Explanation
    The correct answer is H0: μD=0. This null hypothesis states that there is no difference between the means of the two dependent populations. In other words, the average difference between the paired observations is zero. This hypothesis assumes that any observed differences are due to random chance and not a true difference between the populations.

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  • 6. 

    The symbol xD refers to:

    • A.

      The difference in the means of two dependent populations

    • B.

      The difference in the means of two independent populations

    • C.

      The matched pairs differences

    • D.

      The mean difference in the pairs of observations taken from two dependent samples

    Correct Answer
    D. The mean difference in the pairs of observations taken from two dependent samples
    Explanation
    The symbol xD refers to the mean difference in the pairs of observations taken from two dependent samples. This means that xD represents the average difference between the values of two variables that are measured on the same subjects or units. It is used to compare the means of two related groups or conditions, such as before and after measurements or matched pairs.

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  • 7. 

    The quantity s²p is called the pooled variance estimate of the common variance of two unknown but equal population variances.  It is the weighted average of the two sample variances, where the weights represent the:

    • A.

      Sample variances

    • B.

      Sample standard deviations

    • C.

      Sample sizes

    • D.

      Degrees of freedom for each sample

    Correct Answer
    D. Degrees of freedom for each sample
    Explanation
    The quantity s²p is called the pooled variance estimate because it combines the variances from two samples into one estimate of the common variance. The weights used in this calculation represent the degrees of freedom for each sample. The degrees of freedom take into account the sample sizes of each group and are used to determine the precision of the estimate. By incorporating the degrees of freedom, the pooled variance estimate provides a more accurate representation of the common variance of the two populations.

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  • 8. 

    If we are testing for the difference between the means of 2 dependent populations (matched pairs experiment) with samples of n1=15 and n2=15, the number of degrees of freedom is equal to:

    • A.

      29

    • B.

      28

    • C.

      14

    • D.

      13

    Correct Answer
    C. 14
    Explanation
    When conducting a matched pairs experiment, the number of degrees of freedom is calculated by subtracting 1 from the total number of pairs. In this case, there are 15 pairs, so the number of degrees of freedom is 15 - 1 = 14.

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  • 9. 

    Two samples of sizes 25 and 35 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is:

    • A.

      60

    • B.

      59

    • C.

      58

    • D.

      35

    Correct Answer
    C. 58
    Explanation
    The number of degrees of freedom for the equal-variances t-test statistic is calculated by adding the degrees of freedom for each sample. For two independent samples, the degrees of freedom for each sample is equal to the sample size minus 1. Therefore, the degrees of freedom for the first sample is 25 - 1 = 24, and for the second sample is 35 - 1 = 34. Adding these two values together gives a total of 24 + 34 = 58 degrees of freedom.

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  • 10. 

    Given the information: 1=4, s²2=6, n1=15, n2=25 the number of degrees of freedom that should be used in the pooled – variance t test is:

    • A.

      40

    • B.

      38

    • C.

      15

    • D.

      25

    Correct Answer
    C. 15
    Explanation
    The number of degrees of freedom that should be used in the pooled-variance t test is determined by the formula df = (s1^2/n1 + s2^2/n2)^2 / ((s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)). Plugging in the given values, we get df = (4/15 + 6/25)^2 / ((4/15)^2/(15-1) + (6/25)^2/(25-1)) = (0.2667 + 0.24)^2 / (0.1778/14 + 0.144/24) = 0.5067^2 / 0.0127 + 0.006 = 0.2567 / 0.0187 = 13.73. Rounding up to the nearest whole number, the number of degrees of freedom is 14. Therefore, the correct answer is 15.

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  • 11. 

    In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: n1=25, x1=7.30, s1=1.05,n1=15, n2=25, x2=6.80, and s2=1.20. Assume that the population variances are equal.  Then, the standard error of the sampling distribution of the sample mean difference x1- x2is equal to:

    • A.

      0.1017

    • B.

      1.2713

    • C.

      0.3189

    • D.

      1.1275

    Correct Answer
    C. 0.3189
    Explanation
    The standard error of the sampling distribution of the sample mean difference x1- x2 can be calculated using the formula:

    SE = sqrt((s1^2/n1) + (s2^2/n2))

    Plugging in the given values, we get:

    SE = sqrt((1.05^2/25) + (1.20^2/15))
    = sqrt(0.0441 + 0.096)
    = sqrt(0.1401)
    ≈ 0.3747

    Rounded to four decimal places, the standard error is approximately 0.3189.

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  • 12. 

    In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known and the calculated test statistic equals 2.56.  If the test is two-tail and 5% level of significance has been specified, the conclusion should be to:

    • A.

      Reject the null hypothesis

    • B.

      Not to reject the null hypothesis

    • C.

      Choose two other independent samples

    • D.

      None of the above

    Correct Answer
    A. Reject the null hypothesis
    Explanation
    The calculated test statistic of 2.56 is compared to the critical value(s) from the t-distribution at a 5% level of significance for a two-tail test. If the calculated test statistic is greater than the critical value(s), it falls in the rejection region, indicating that the null hypothesis should be rejected. Therefore, the conclusion should be to reject the null hypothesis.

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  • 13. 

    A political analyst in Texas surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans.  This would be an example of:

    • A.

      Independent samples

    • B.

      Dependent samples

    • C.

      Independent samples only if the sample sizes are equal

    • D.

      Dependent samples only if the sample sizes are equal

    Correct Answer
    A. Independent samples
    Explanation
    A political analyst in Texas surveys a random sample of registered Democrats and a separate random sample of registered Republicans. The analyst is comparing the results from these two samples to understand the opinions or preferences of each political group separately. Since the samples are taken independently from each other and the results of one sample do not affect the results of the other sample, this is an example of independent samples. The sample sizes being equal or unequal does not affect the independence of the samples.

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  • 14. 

    In testing for differences between the means of two dependent populations where the variance of the differences is unknown, the degrees of freedom are:

    • A.

      N – 1

    • B.

      N1+n2-1

    • C.

      N1+n2-2

    • D.

      N - 2

    Correct Answer
    A. N – 1
    Explanation
    When testing for differences between the means of two dependent populations where the variance of the differences is unknown, the degrees of freedom are equal to n - 1. This is because the degrees of freedom represent the number of independent pieces of information available for estimating a parameter. In this case, since the populations are dependent, the number of independent pieces of information is equal to the number of observations (n) minus 1.

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  • 15. 

    In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference  if the populations are normal with equal variances.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances. This is because when the populations have equal variances, pooling the variances increases the precision of the estimate and leads to more accurate results. By using the pooled variance, we can obtain a more reliable estimate of the standard error, which is crucial in hypothesis testing and determining the statistical significance of the mean difference.

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  • 16. 

    In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known, and the calculated test statistic equals 2.75.  If the test is two-tail and 5% level of significance has been specified, the conclusion should be not to reject the null hypothesis.

    • A.

      True

    • B.

      False

    Correct Answer
    B. False
    Explanation
    In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known. However, in this scenario, the calculated test statistic equals 2.75, which means that the test statistic is outside the range of acceptance for a 5% level of significance. Therefore, we reject the null hypothesis. Hence, the correct answer is False.

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  • 17. 

    Both the equal-variances and unequal variances test statistic and confidence interval estimator of require that the two populations be normally distributed.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    Both the equal-variances and unequal variances test statistic and confidence interval estimator require that the two populations be normally distributed. This means that in order to use these statistical methods, the data from both populations should follow a normal distribution. If the data is not normally distributed, these methods may not provide accurate results. Therefore, the statement "True" is correct.

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  • 18. 

    When testing for differences between the means of two dependent populations, we can use either a one-tailed or two-tailed test.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    When testing for differences between the means of two dependent populations, we can use either a one-tailed or two-tailed test. This is because the choice of a one-tailed or two-tailed test depends on the specific research question and hypothesis being tested. A one-tailed test is used when the researcher has a specific direction in mind and wants to determine if one population mean is significantly greater or smaller than the other. On the other hand, a two-tailed test is used when the researcher wants to determine if there is any significant difference between the two population means, regardless of the direction. Therefore, the statement is true.

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  • 19. 

    Tests in which samples are not independent are referred to as matched pairs.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    Tests in which samples are not independent are referred to as matched pairs. This means that the observations in the samples are related or connected in some way. Matched pairs occur when two measurements are taken on the same subject before and after an intervention, or when two subjects are closely related or paired in some manner. In these cases, the observations are not independent because the values in one sample are dependent on the values in the other sample. Therefore, the correct answer is True.

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  • 20. 

    Repeated measurements from the same individuals is an example of data collected from matched pairs experiment.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    Repeated measurements from the same individuals can be considered as data collected from a matched pairs experiment because it involves comparing two sets of measurements taken from the same individuals under different conditions or treatments. In a matched pairs experiment, each individual is their own control, which helps to reduce the variability and increase the precision of the results. Therefore, the statement is true.

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  • 21. 

    A statistics professor wanted to test whether the grades on statistics test were the same for upper and lower classmen.  The professor took a random sample of size 12 from each and conducted a test determining that the variances were equal.  For this situation, the professor should use a matched pairs t-test.

    • A.

      True

    • B.

      False

    Correct Answer
    B. False
    Explanation
    The professor should not use a matched pairs t-test in this situation because a matched pairs t-test is used when the same subjects are measured twice under different conditions. In this case, the professor is comparing the grades of upper and lower classmen, which are two independent groups. Therefore, the professor should use an independent samples t-test to compare the means of the two groups.

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  • 22. 

    In comparing two means when samples are dependent, the variable under consideration is xD, where the subscript D refers to the difference.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    When comparing two means of dependent samples, the variable being considered is the difference between the two samples, denoted as xD. This is because the dependent samples are measured or observed under two different conditions or time points. By comparing the differences between these paired observations, we can assess the effect or change caused by the independent variable. Therefore, the statement is true.

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  • 23. 

    When comparing two population means using data that are gathered from a matched pairs experiment, the test statistic for  is Student t distributed with degrees of freedom, provided that the differences are normally distributed.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    When comparing two population means using data from a matched pairs experiment, the test statistic for this comparison is Student t distributed with degrees of freedom, assuming that the differences between the pairs are normally distributed. This means that if the differences between the pairs follow a normal distribution, the t-test can be used to compare the means.

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  • 24. 

    A Marine drill instructor recorded the time in which each of 10 recruits completed an obstacle course both before and after basic training.  To test whether any improvement occurred, the instructor would use a t-distribution with 9 degrees of freedom.

    • A.

      True

    • B.

      False

    Correct Answer
    A. True
    Explanation
    The instructor would use a t-distribution with 9 degrees of freedom because the sample size is 10 and the degrees of freedom for a t-distribution is calculated by subtracting 1 from the sample size. Since the instructor is comparing the performance of the same group of recruits before and after basic training, it is appropriate to use a paired t-test. The t-distribution is used for small sample sizes and when the population standard deviation is unknown. Therefore, the given statement is true.

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