Question

Chapter 10) Several factors influence the value obtained for the independent-measures t statistic. Some factors affect the numerator of the t statistic and others influence the size of the estimated standard error in the denominator. For each of the following, indicate whether the factor influences the numerator or denominator of the t statistic and determine whether the effect would be to increase the value of t (farther from zero) or decrease the value of t (closer to zero). In each case, assume all other factors remain constant. a. Decrease the size of the sample variances (explain the logic behind your answer). b. Decrease the difference between the two samples (explain the logic behind your answer). c. Decrease the size of the two samples (explain the logic behind your answer). Please give an example equation for A, B and C.

Answer 1

Solution:

The independent measure t statistic is given as follows:

Where, x̄ and ȳ are sample means, n_{​​​​​​1} and n_{​​​​​​2} are sample sizes and S^{​​​​​​2}_{​​​​​pooled} is pooled variance.

s_{​​​​​​1}^{​​​​​2} and s_{​​​​​2}² are sample variances.

a) Since, sample variances are in the denominator of test statistic, therefore it would affect the denominator of test statistic. When we decrease the sample variances, the value of the S^{​​​​​​2}_{​​​​​pooled} decreases and when the S^{​​​​​​2}_{​​​​​pooled} decreases the denominator of the test statistic decreases. With the decrease in the denominator of the test statistic the value of the independent measure t test statistic increases.

For example suppose initially we have following values:

x̄ = 19, ȳ = 15 , n_{​​​​​​1} = 16, n₂ = 16, s_{​​​​​​1}^{​​​​​2} = 9 and s_{​​​​​2}² = 7

Now we decrease the sample variances. The new sample variances are, s_{​​​​​​1}^{​​​​​2} = 7 and s_{​​​​​2}² = 5

From the above example we can see that on decreasing the sample variances, the denominator of the test statistic decreases and the value of the test statistic increases.

b) Since, difference between two sample means is in the numerator of the test statistic, therefore decreasing the difference between two samples would affect the numerator of the test statistic. On decreasing the difference between two sample means the value of the numerator decreases which results the decrease in the value of the test statistic.

For example suppose initially we have following values:

x̄ = 19, ȳ = 15 , n_{​​​​​​1} = 16, n₂ = 16, s_{​​​​​​1}^{​​​​​2} = 9 and s_{​​​​​2}² = 7

Now we decrease the difference between two sample means. The new sample means are, x̄ = 19, ȳ = 17

From the above example we can see that decreasing the difference between two sample means would decrease the numerator of the test statistic which results the decrease in the value of the test statistic.

c) Decreasing the size of the samples would affect the denominator of the test statistic. Decreasing sample size would increase the value of the denominator which results the decrease in the value of the test statistic.

For example suppose initially we have following values:

x̄ = 19, ȳ = 15 , n_{​​​​​​1} = 16, n₂ = 16, s_{​​​​​​1}^{​​​​​2} = 9 and s_{​​​​​2}² = 7

Now we decrease the size of two samples. The new sample sizes are, n_{​​​​​​1} = 12, n₂ = 13

From the above example we can see that decreasing sample size would increase the value of the denominator which results the decrease in the value of the test statistic.