Exercise 1

Consider a random sample of size \(n = 50\). The sample mean and standard deviation are:

Use this sample to test \(H_0\colon \ \mu = 4 \ \text{ vs } \ H_1\colon \ \mu > 4\).

Report:

Solution

  • Test statistic: \(z = 2.36\)
  • Crictical value: \(1.645\)
    • Reject if \(z > 1.645\)
  • Decision: Reject \(H_0\)

Exercise 2

Consider a random sample of size \(n = 100\). The sample mean and standard deviation are:

Use this sample to test \(H_0\colon \ \mu = 0 \ \text{ vs } \ H_1\colon \ \mu \neq 0\).

Report:

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 \(X\):

Sample 2, from Population \(Y\):

Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).

Report:

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 \(X\):

Sample 2, from Population \(Y\):

Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).

Report:

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:

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:

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 \(X\):

Sample 2, from Population \(Y\):

Use these samples to test \(H_0\colon \ p_x = p_y \ \text{ vs } \ H_1\colon \ p_x \neq p_y\).

Report:

Solution

  • Test statistic: \(z = -1.39\)
    • \(\hat{p} = 0.75\)
  • P-value: \(0.1646\)
  • Decision: Fail to reject \(H_0\)

Exercise 8

Consider two independent random samples from dichotomous populations.

Sample 1, from Population \(X\):

Sample 2, from Population \(Y\):

Use these samples to test \(H_0\colon \ p_x = p_y \ \text{ vs } \ H_1\colon \ p_x \neq p_y\).

Report:

Solution

  • Test statistic: \(z = -1.96\)
    • \(\hat{p} = 0.47\)
  • 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:

Use this sample to test \(H_0\colon \ \mu = 4 \ \text{ vs } \ H_1\colon \ \mu > 4\).

Report:

Solution

  • Test statistic: \(t = 1.76\)
    • \(df = 11\)
  • Crictical value: \(1.796\)
    • Reject if \(t > 1.796\)
  • 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:

Use this sample to test \(H_0\colon \ \mu = 0 \ \text{ vs } \ H_1\colon \ \mu < 0\).

Report:

Solution

  • Test statistic: \(t = -1.13\)
    • \(df = 7\)
  • Crictical value: \(-1.415\)
    • Reject if \(t < -1.415\)
  • 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:

Use this sample to test \(H_0\colon \ \sigma = 2 \ \text{ vs } \ H_1\colon \ \sigma > 2\).

Report:

Solution

  • Test statistic: \(X^2 = 14.5475\)
    • \(df = 11\)
  • 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:

Use this sample to test \(H_0\colon \ \sigma = 5 \ \text{ vs } \ H_1\colon \ \sigma > 5\).

Report:

Solution

  • Test statistic: \(X^2 = 26.3424\)
    • \(df = 21\)
  • 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 \(X\):

Sample 2, from Population \(Y\):

Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).

Report:

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 \(X\):

Sample 2, from Population \(Y\):

Use these samples to test \(H_0\colon \ \mu_x = \mu_y \ \text{ vs } \ H_1\colon \ \mu_x \neq \mu_y\).

Report:

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 \(H_0\)