A 95% confidence interval for u is computed to be (1.50, 3.05). For each of the following hypotheses, state whether will be rejected at 0.05 level.

Question

Hv0 : u = 4 versus Hv1 : u does not = 4
Does the 95% confidence level contain uvo? Is Hv0 rejected?

Answer 1

To determine whether the null hypothesis \(H_{0}: \mu = 4\) will be rejected at the 0.05 significance level, we examine the confidence interval (CI) provided, which is (1.50, 3.05).

1. Does the 95% confidence interval contain \(\mu_0\)? The value under consideration is \(\mu_0 = 4\). Since \(\mu_0 = 4\) is outside the confidence interval (1.50, 3.05), this means that the interval does not contain the value of the null hypothesis.

2. Is \(H_{0}\) rejected? Since the value of 4 is not included in the 95% confidence interval, we reject the null hypothesis \(H_{0}\) at the 0.05 significance level.

In summary:

• The 95% confidence interval does not contain \(\mu_0 = 4\).
• The null hypothesis \(H_{0}: \mu = 4\) will be rejected at the 0.05 level.

Answer 2

Hv0 : u = 2versus Hv1 : u does not = 2
Does the 95% confidence level contain uvo? Is Hv0 rejected?

Answer 3

We are examining the hypothesis:

• \(H_{0}: \mu = 2\) (null hypothesis)
• \(H_{1}: \mu \neq 2\) (alternative hypothesis)

Given that the 95% confidence interval for \(\mu\) is (1.50, 3.05):

1. Does the 95% confidence interval contain \(\mu_0\)?

   • The value under consideration is \(\mu_0 = 2\). Since \(2\) is within the confidence interval (1.50, 3.05), this means that the interval does contain the value of the null hypothesis.

2. Is \(H_{0}\) rejected?

   • Since the value of \(2\) is included in the 95% confidence interval, we do not reject the null hypothesis \(H_{0}\) at the 0.05 significance level.

In summary:

• The 95% confidence interval does contain \(\mu_0 = 2\).
• The null hypothesis \(H_{0}: \mu = 2\) will not be rejected at the 0.05 level.

Answer 4

Hv0 : u = 2.8 versus Hv1 : u does not = 2.8
Does the 95% confidence level contain uvo? Is Hv0 rejected?

Answer 5

We are examining the hypotheses:

• \(H_{0}: \mu = 2.8\) (null hypothesis)
• \(H_{1}: \mu \neq 2.8\) (alternative hypothesis)

Given the 95% confidence interval for \(\mu\) is (1.50, 3.05):

1. Does the 95% confidence interval contain \(\mu_0\)?

   • The value under consideration is \(\mu_0 = 2.8\). Since \(2.8\) is within the confidence interval (1.50, 3.05), this means that the interval does contain the value of the null hypothesis.

2. Is \(H_{0}\) rejected?

   • Since the value of \(2.8\) is included in the 95% confidence interval, we do not reject the null hypothesis \(H_{0}\) at the 0.05 significance level.

In summary:

• The 95% confidence interval does contain \(\mu_0 = 2.8\).
• The null hypothesis \(H_{0}: \mu = 2.8\) will not be rejected at the 0.05 level.

Answer 6

Hv0 : u = 2.3 versus Hv1 : u does not = 2.3
Does the 95% confidence level contain uvo? Is Hv0 rejected?

Answer 7

We are examining the hypotheses:

• \(H_{0}: \mu = 2.3\) (null hypothesis)
• \(H_{1}: \mu \neq 2.3\) (alternative hypothesis)

Given the 95% confidence interval for \(\mu\) is (1.50, 3.05):

1. Does the 95% confidence interval contain \(\mu_0\)?

   • The value under consideration is \(\mu_0 = 2.3\). Since \(2.3\) is within the confidence interval (1.50, 3.05), this means that the interval does contain the value of the null hypothesis.

2. Is \(H_{0}\) rejected?

   • Since the value of \(2.3\) is included in the 95% confidence interval, we do not reject the null hypothesis \(H_{0}\) at the 0.05 significance level.

In summary:

• The 95% confidence interval does contain \(\mu_0 = 2.3\).
• The null hypothesis \(H_{0}: \mu = 2.3\) will not be rejected at the 0.05 level.