Question

Simulation.In this exercise, you are to perform a compute simulation to illustrate the distribution of the pooled $t$-statistic, given in Key Fact $10.2$on page $407$.

a. Simulate $1000$random samples of size $4$from a normally distributed variable with a mean of $100$and a standard deviation of $16$. Then obtain the sample mean and sample standard deviation of each of the $1000$samples.

b. Simulate $1000$random samples of size $3$from a normally distributed variable with a mean of $110$and a standard deviation of $16$Then obtain the sample mean and sample standard deviation of each of the$1000$samples.

c. Determine the value of the pooled $t$-statistic for each of the $1000$pairs of samples obtained in parts $\left(a\right)$and $\left(b\right)$.

d. Obtain a histogram of the $1000$values found in part $\left(c\right)$.

e. Theoretically, what is the distribution of all possible values of the pooled $t$-statistic?

f. Compare your results from parts$\left(d\right)$and$\left(e\right)$.

Short answer

Part $\left(a\right)$: The mean and standard deviation of each of the $1000$samples is, $t=\frac{\left({\displaystyle {\stackrel{-}{x}}_1-{\stackrel{-}{x}}_2}\right)}{s_P\sqrt{{\displaystyle \frac{1}{n_1}+\frac{1}{n_2}}}}$.

Part $\left(b\right)$: The mean and standard deviation of each of the $1000$samples is, $t=\frac{\left({\displaystyle {\stackrel{-}{x}}_1-{\stackrel{-}{x}}_2}\right)}{s_P\sqrt{{\displaystyle \frac{1}{4}+\frac{1}{3}}}}$.

Part $\left(c\right)$: The value of the pooled $t$-statistic for each of the $1000$pairs of samples obtained in parts $\left(a\right)$and $\left(b\right)$is, $96.062$.

Part $\left(d\right)$: A histogram for the pooled $t$-statistic:

Part $\left(e\right)$: The distribution of all possible values of the pooled $t$-statistic is, $96.062$.

Part$\left(f\right)$: We observed that the pooled $t$-statistics for $1000$pairs of random samples satisfying normality condition, which was shown in the histogram.

Step-by-step solution

Part a Step 1. Given information

We need to simulate$1000$random samples of size$4$from a normally distributed variable with a mean of$100$and a standard deviation of$16$.

Part a Step 2. Now we need to obtain the mean and standard deviation of each of the 1000 samples are shown in the last two columns of the table.

$t=\frac{\left({{\displaystyle \stackrel{-}{x}}}_1-{\stackrel{-}{x}}_2\right)}{s_P\sqrt{{\displaystyle \frac{1}{n_1}}+{\displaystyle \frac{1}{n_2}}}}$.

Part b Step 1. Here, we need to simulate 1000 random samples of size 4 from a normally distributed variable.

Also, the mean is, $100$and the standard deviation is, $16$. We need to obtain the mean and standard deviation of each of the $1000$samples are shown in the last two columns of the table.

33.

$t=\frac{\left({\displaystyle {\stackrel{-}{x}}_1-{\stackrel{-}{x}}_2}\right)}{s_P\sqrt{{\displaystyle \frac{1}{4}+\frac{1}{3}}}}$.

Part c Step 1. Pooled t-statistic is, 

$$\begin{gathered} t=\frac{\left({\displaystyle {\stackrel{-}{x}}_1-{\stackrel{-}{x}}_2}\right)}{s_P\sqrt{{\displaystyle \frac{7}{12}}}} \\ {s}_P=\sqrt{\frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}} \\ =\sqrt{\frac{\left(4-1\right)s_1^2+\left(3-1\right)s_2^2}{4+3-2}} \\ {s}_P=\sqrt{\frac{3s_1^2+2s_2^2}{5}} \\ =\sqrt{\frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}} \end{gathered}$$

Part c Step 2. we obtain the pooled t-statistic for each of the 1000 pairs of the samples obtained in parts a and b which are shown:

$$\begin{gathered} s_P=\sqrt{\frac{\left(4-1\right){\left(89.1\right)}^2+\left(3-1\right){\left(105.648\right)}^2}{4+3-2}} \\ =96.062 \end{gathered}$$

Part d Step 1. Let us draw a histogram for the pooled t-statistic:

Part e Step 1. Theoretically, the distribution of all the possible values of the pooled t-statistic is obtained by,

$$\begin{gathered} s_P=\frac{\left(89.1-105.648\right)}{\left(96.062\right)\sqrt{{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{3}}}} \\ =-0.22556 \\ =96.062 \end{gathered}$$

Now apply the same procedure for the remaining$999$pairs of values.

Part f Step 1. Now compare the results from the part d and e. 

We observed that the pooled $t$-statistics for $1000$pairs of random samples satisfying normality condition, which was shown in the histogram.