STAT 3202: Practice 06

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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 \(\alpha = 0.05\).

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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 \(\alpha = 0.01\).

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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 \(\alpha = 0.05\).

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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 \(\alpha = 0.05\).

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Exercise 5

Consider a random sample of size \(n = 50\) from a dichotomous population. The sample proportion of the “success” class is

• \(\hat{p} = 0.58\)

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 \(\alpha = 0.01\).

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Exercise 6

Consider a random sample of size \(n = 100\) from a dichotomous population. The sample proportion of the “success” class is

• \(\hat{p} = 0.81\)

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 \(\alpha = 0.05\).

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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 \(\alpha = 0.10\).

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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 \(\alpha = 0.01\).

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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 \(\alpha = 0.05\).

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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 \(\alpha = 0.10\).

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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 \(\alpha = 0.05\).

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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 \(\alpha = 0.01\).

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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 \(\alpha = 0.05\).

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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 \(\alpha = 0.01\).

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