Chapter 9, Problem 16CAP

1.

To determine

Identify whether the test statistic in a two independent sample t test increases, decreases, or have no effect when sample size is increased.

Answer to Problem 16CAP

The test statistic in a two independent sample t test increases when sample size is increased.

Explanation of Solution

Justification: The formula of test statistic for one-sample t test is,

$t_{\text{obt}}=\frac{\left(M_1-M_2\right)-\left({\mu }_1-{\mu }_2\right)}{s_{M_1-M_2}}$

In the formula, $M_1$ is first sample mean, $M_2$ is second sample mean, ${\mu }_1$ is first population mean, ${\mu }_2$ is second population mean, $s_{M_1-M_2}$ is estimated standard error of difference given as,

$s_{M_1-M_2}=\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}$ and $s_p^2=\frac{s_1^2\left(d{f}_1\right)+s_2^2\left(d{f}_2\right)}{d{f}_1+d{f}_2}$.

In the formula, $s_1^2$ is the first sample standard deviation, $s_2^2$ is the second sample standard deviation, $d{f}_1$ are the degrees of freedom for first sample and $d{f}_2$ are the degrees of freedom for second sample.

The value of sample size is in the denominator of the standard error; if the sample size is increased then standard error would decrease, because standard error and sample size are inversely proportional.

Hence, the test statistic in a two independent sample t test increases when sample size is increased.

2.

To determine

Identify whether the test statistic in a two independent sample t test increases, decreases, or have no effect when level of significance is reduced from 0.05 to 0.01.

Answer to Problem 16CAP

The test statistic in a two independent sample t test has no effect when level of significance is reduced from 0.05 to 0.01

Explanation of Solution

Justification: The formula of test statistic for one-sample t test is,

$t_{\text{obt}}=\frac{\left(M_1-M_2\right)-\left({\mu }_1-{\mu }_2\right)}{s_{M_1-M_2}}$

In the formula, $M_1$ is first sample mean, $M_2$ is second sample mean, ${\mu }_1$ is first population mean, ${\mu }_2$ is second population mean, $s_{M_1-M_2}$ is estimated standard error of difference.

The level of significance value is not used in the calculation of the test statistic value in two independent sample t test. Whether the level of significance is increased or decreased, it does not have any effect on test statistic.

Hence, the test statistic in a two independent sample t test has no effect when level of significance is reduced from 0.05 to 0.01.

3.

To determine

Identify whether the test statistic in a two independent sample t test increases, decreases, or have no effect when pooled sample variance is doubled.

Answer to Problem 16CAP

The test statistic in a two independent sample t test decreases when pooled sample variance is doubled.

Explanation of Solution

Justification: The formula of test statistic for one-sample t test is,

$t_{\text{obt}}=\frac{\left(M_1-M_2\right)-\left({\mu }_1-{\mu }_2\right)}{s_{M_1-M_2}}$

In the formula, $M_1$ is first sample mean, $M_2$ is second sample mean, ${\mu }_1$ is first population mean, ${\mu }_2$ is second population mean, $s_{M_1-M_2}$ is estimated standard error of difference given as,

$s_{M_1-M_2}=\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}$ and $s_p^2=\frac{s_1^2\left(d{f}_1\right)+s_2^2\left(d{f}_2\right)}{d{f}_1+d{f}_2}$.

In the formula, $s_1^2$ is the first sample standard deviation, $s_2^2$ is the second sample standard deviation, $d{f}_1$ are the degrees of freedom for first sample and $d{f}_2$ are the degrees of freedom for second sample.

If the value of pooled sample variance is doubled then pooled sample standard deviation is also doubled.

The value of pooled sample standard deviation is in the numerator of the standard error, if the pooled sample standard deviation is doubled then standard error would increase, because standard error and pooled sample standard deviation are directly proportional.

The standard error value is in the denominator of test statistic value, if standard error is increased, the test statistic value decreases as standard error and test statistic are inversely proportional to each other.

Hence, the test statistic in a two independent sample t test decreases when pooled sample variance is doubled.