Question

part 2 of 5
step 1. hypotheses set-up:
$h_0:$ $mu_1$-$mu_2$ = 0
, where $mu$s are the population means and the units are $$
$h_a:$ $mu_1$-$mu_2$ < 0
, and the test is left-tail
step 2. the significance level $alpha$ = 5 %
part 3 of 5
step 3. compute the value of the test statistic: select an answer = (round the answer to 3 decimal places)
step 4. testing procedure: (round the answers to 3 decimal places)

cva	pva
left cv is and right cv is	p-value is

enter an integer or decimal number, with 3 decimal places (more..)
question help: video 1 video 2 written example 1

Explanation:

To solve this hypothesis testing problem, we need the sample data (means, standard deviations, sample sizes) for the two populations. Since the data is not provided in the question, we can't compute the test statistic, critical values, or p - value.

Step - by - Step Explanation (if data was provided)

Step 1: Recall the formula for the test statistic in two - sample t - test (assuming equal variances or appropriate conditions)

The formula for the test statistic \(t\) in a two - sample t - test is \(t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\), where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(\mu_1-\mu_2\) is the hypothesized difference in population means (in our case, under \(H_0\), \(\mu_1-\mu_2 = 0\)), \(s_p\) is the pooled standard deviation, and \(n_1\) and \(n_2\) are the sample sizes. The pooled standard deviation \(s_p=\sqrt{\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}}\), where \(s_1\) and \(s_2\) are the sample standard deviations.

Step 2: Calculate the critical values

For a left - tailed test with \(\alpha = 0.05\) and degrees of freedom \(df=n_1 + n_2-2\), we can find the critical value \(t_{\alpha,df}\) from the t - distribution table. If it is a z - test (when sample sizes are large or population standard deviations are known), the critical value \(z_{\alpha}\) for a left - tailed test with \(\alpha=0.05\) is \(z=- 1.645\).

Step 3: Calculate the p - value

The p - value for a left - tailed test is the probability that \(T\leq t_{stat}\) (for t - test) or \(Z\leq z_{stat}\) (for z - test), where \(t_{stat}\) or \(z_{stat}\) is the calculated test statistic. We can find the p - value using the t - distribution or z - distribution table or using statistical software.

Since the necessary sample data (sample means, sample standard deviations, sample sizes) is not provided in the question, we cannot provide the numerical values for the test statistic, critical values, and p - value. If you provide the sample data (e.g., \(\bar{x}_1,\bar{x}_2,s_1,s_2,n_1,n_2\)), we can complete the calculations.

Answer:

To solve this hypothesis testing problem, we need the sample data (means, standard deviations, sample sizes) for the two populations. Since the data is not provided in the question, we can't compute the test statistic, critical values, or p - value.

Step - by - Step Explanation (if data was provided)

Step 1: Recall the formula for the test statistic in two - sample t - test (assuming equal variances or appropriate conditions)

The formula for the test statistic \(t\) in a two - sample t - test is \(t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\), where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(\mu_1-\mu_2\) is the hypothesized difference in population means (in our case, under \(H_0\), \(\mu_1-\mu_2 = 0\)), \(s_p\) is the pooled standard deviation, and \(n_1\) and \(n_2\) are the sample sizes. The pooled standard deviation \(s_p=\sqrt{\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}}\), where \(s_1\) and \(s_2\) are the sample standard deviations.

Step 2: Calculate the critical values

For a left - tailed test with \(\alpha = 0.05\) and degrees of freedom \(df=n_1 + n_2-2\), we can find the critical value \(t_{\alpha,df}\) from the t - distribution table. If it is a z - test (when sample sizes are large or population standard deviations are known), the critical value \(z_{\alpha}\) for a left - tailed test with \(\alpha=0.05\) is \(z=- 1.645\).

Step 3: Calculate the p - value

The p - value for a left - tailed test is the probability that \(T\leq t_{stat}\) (for t - test) or \(Z\leq z_{stat}\) (for z - test), where \(t_{stat}\) or \(z_{stat}\) is the calculated test statistic. We can find the p - value using the t - distribution or z - distribution table or using statistical software.

Since the necessary sample data (sample means, sample standard deviations, sample sizes) is not provided in the question, we cannot provide the numerical values for the test statistic, critical values, and p - value. If you provide the sample data (e.g., \(\bar{x}_1,\bar{x}_2,s_1,s_2,n_1,n_2\)), we can complete the calculations.