Exercise 1
Consider a random sample of size \(n = 50\). The sample mean and standard deviation are:
- \(\bar{x} = 5\)
- \(s = 3\)
Use this sample to test \(H_0\colon \ \mu = 4 \ \text{ vs } \ H_1\colon \ \mu > 4\).
Report:
- The test statistic
- The crictical value when \(\alpha = 0.05\).
- A decision when \(\alpha = 0.05\).
Solution
- Test statistic: \(z = 2.36\)
- Crictical value: \(1.645\)
- Decision: Reject \(H_0\)
Exercise 2
Consider a random sample of size \(n = 100\). The sample mean and standard deviation are:
- \(\bar{x} = -0.5\)
- \(s = 2\)
Use this sample to test \(H_0\colon \ \mu = 0 \ \text{ vs } \ H_1\colon \ \mu \neq 0\).
Report:
- The test statistic
- The p-value
- A decision when \(\alpha = 0.01\).
Solution
- Test statistic: \(z = -2.5\)
- P-value: \(0.0124\)
- Decision: Fail to reject \(H_0\)
Exercise 3
Consider two independent random samples.
Sample 1, from Population \(X\):
- \(n_x = 50\)
- \(\bar{x} = 5\)
- \(s_x = 2\)
Sample 2, from Population \(Y\):
- \(n_y = 45\)
- \(\bar{y} = 6\)
- \(s_y = 3\)
Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).
Report:
- The test statistic
- The p-value
- A decision when \(\alpha = 0.05\).
Solution
- Test statistic: \(z = -1.89\)
- P-value: \(0.0588\)
- Decision: Fail to reject \(H_0\)
Exercise 4
Consider two independent random samples.
Sample 1, from Population \(X\):
- \(n_x = 75\)
- \(\bar{x} = 13\)
- \(s_x = 5\)
Sample 2, from Population \(Y\):
- \(n_y = 100\)
- \(\bar{y} = 11\)
- \(s_y = 6\)
Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).
Report:
- The test statistic
- The crictical values when \(\alpha = 0.05\).
- A decision when \(\alpha = 0.05\).
Solution
- Test statistic: \(z = 2.40\)
- Crictical values: \(-1.960, 1.960\)
- Reject if \(z > 1.960\)
- Reject if \(z < -1.960\)
- Decision: Reject \(H_0\)
Exercise 5
Consider a random sample of size \(n = 50\) from a dichotomous population. The sample proportion of the “success” class is
Use this sample to test \(H_0\colon \ p = 0.50 \ \text{ vs } \ H_1\colon \ p \neq 0.50\).
Report:
- The test statistic
- The crictical value when \(\alpha = 0.01\).
- A decision when \(\alpha = 0.01\).
Solution
- Test statistic: \(z = 1.13\)
- Crictical values: \(-2.576, 2.576\)
- Reject if \(z > 2.576\)
- Reject if \(z < -2.576\)
- Decision: Fail to reject \(H_0\)
Exercise 6
Consider a random sample of size \(n = 100\) from a dichotomous population. The sample proportion of the “success” class is
Use this sample to test \(H_0\colon \ p = 0.70 \ \text{ vs } \ H_1\colon \ p > 0.70\).
Report:
- The test statistic
- The p-value
- A decision when \(\alpha = 0.05\).
Solution
- Test statistic: \(z = 2.40\)
- P-value: \(0.0082\)
- Decision: Reject \(H_0\)
Exercise 7
Consider two independent random samples from dichotomous populations.
Sample 1, from Population \(X\):
- \(n_x = 80\)
- \(\hat{p}_x = 0.70\)
Sample 2, from Population \(Y\):
- \(n_y = 90\)
- \(\hat{p}_y = 0.79\)
Use these samples to test \(H_0\colon \ p_x = p_y \ \text{ vs } \ H_1\colon \ p_x \neq p_y\).
Report:
- The test statistic
- The p-value
- A decision when \(\alpha = 0.10\).
Solution
- Test statistic: \(z = -1.39\)
- P-value: \(0.1646\)
- Decision: Fail to reject \(H_0\)
Exercise 8
Consider two independent random samples from dichotomous populations.
Sample 1, from Population \(X\):
- \(n_x = 100\)
- \(\hat{p}_x = 0.39\)
Sample 2, from Population \(Y\):
- \(n_y = 200\)
- \(\hat{p}_y = 0.51\)
Use these samples to test \(H_0\colon \ p_x = p_y \ \text{ vs } \ H_1\colon \ p_x \neq p_y\).
Report:
- The test statistic
- The p-value
- A decision when \(\alpha = 0.01\).
Solution
- Test statistic: \(z = -1.96\)
- P-value: \(0.05\)
- Decision: Fail to reject \(H_0\)
Exercise 9
Consider a random sample of size \(n = 12\) from a population that is assumed to be normal. The sample mean and standard deviation are:
- \(\bar{x} = 5\)
- \(s = 2\)
Use this sample to test \(H_0\colon \ \mu = 4 \ \text{ vs } \ H_1\colon \ \mu > 4\).
Report:
- The test statistic
- The crictical value when \(\alpha = 0.05\).
- A decision when \(\alpha = 0.05\).
Solution
- Test statistic: \(t = 1.76\)
- Crictical value: \(1.796\)
- Decision: Fail to reject \(H_0\)
Exercise 10
Consider a random sample of size \(n = 8\) from a population that is assumed to be normal. The sample mean and standard deviation are:
- \(\bar{x} = -1.2\)
- \(s = 3\)
Use this sample to test \(H_0\colon \ \mu = 0 \ \text{ vs } \ H_1\colon \ \mu < 0\).
Report:
- The test statistic
- The crictical value when \(\alpha = 0.10\).
- A decision when \(\alpha = 0.10\).
Solution
- Test statistic: \(t = -1.13\)
- Crictical value: \(-1.415\)
- Decision: Fail to reject \(H_0\)
Exercise 11
Consider a random sample of size \(n = 12\) from a population that is assumed to be normal. The sample mean and standard deviation are:
- \(\bar{x} = 5\)
- \(s = 2.3\)
Use this sample to test \(H_0\colon \ \sigma = 2 \ \text{ vs } \ H_1\colon \ \sigma > 2\).
Report:
- The test statistic
- The crictical value when \(\alpha = 0.05\).
- A decision when \(\alpha = 0.05\).
Solution
- Test statistic: \(X^2 = 14.5475\)
- Crictical value: \(19.6751\)
- Reject if \(X^2 > 19.6751\)
- Decision: Fail to reject \(H_0\)
Exercise 12
Consider a random sample of size \(n = 22\) from a population that is assumed to be normal. The sample mean and standard deviation are:
- \(\bar{x} = 7\)
- \(s = 5.6\)
Use this sample to test \(H_0\colon \ \sigma = 5 \ \text{ vs } \ H_1\colon \ \sigma > 5\).
Report:
- The test statistic
- The crictical value when \(\alpha = 0.01\).
- A decision when \(\alpha = 0.01\).
Solution
- Test statistic: \(X^2 = 26.3424\)
- Crictical value: \(38.9321\)
- Reject if \(X^2 > 38.9321\)
- Decision: Fail to reject \(H_0\)
Exercise 13
Consider two independent random samples. Assume both populations are normal and that their variances are equal.
Sample 1, from Population \(X\):
- \(n_x = 10\)
- \(\bar{x} = 5\)
- \(s_x = 2\)
Sample 2, from Population \(Y\):
- \(n_y = 12\)
- \(\bar{y} = 6\)
- \(s_y = 1.5\)
Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).
Report:
- The test statistic
- The crictical values when \(\alpha = 0.05\).
- A decision when \(\alpha = 0.05\).
Solution
- Test statistic: \(t = -2.68\)
- \(s_p = \sqrt{3.0375}\)
- \(df = 20\)
- Crictical values: \(-2.086, 2.086\)
- Reject if \(t > 2.086\)
- Reject if \(t < -2.086\)
- Decision: Reject \(H_0\)
Exercise 14
Consider two independent random samples. Assume both populations are normal and that their variances are equal.
Sample 1, from Population \(X\):
- \(n_x = 14\)
- \(\bar{x} = 48,530\)
- \(s_x = 780\)
Sample 2, from Population \(Y\):
- \(n_y = 11\)
- \(\bar{y} = 47,620\)
- \(s_y = 750\)
Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).
Report:
- The test statistic
- The crictical values when \(\alpha = 0.01\).
- A decision when \(\alpha = 0.01\).
Solution
- Test statistic: \(t = 2.944\)
- \(s_p = 767.1\)
- \(df = 23\)
- Crictical values: \(-2.807, 2.807\)
- Reject if \(t > 2.807\)
- Reject if \(t < -2.807\)
- Decision: Reject \(H_0\)