Question

a researcher is going to perform a two - tailed hypothesis test. the test statistic will follow a standard normal distribution. answer parts (a) and (b) below. you may use the tool below in the scratch work area to help you. your work with the tool will not be graded. (a) the researcher might choose 0.01, 0.05, or 0.10 for the level of significance for the two - tailed test. for each potential choice for the level of significance, find the critical values. round your answers to three decimal places. critical values at α = 0.01: and critical values at α = 0.05: and critical values at α = 0.10: and (b) after choosing the level of significance, the researcher takes a sample and finds that the value of the test statistic is z = 2.497. for which of the levels of significance would the null hypothesis be rejected? choose all that apply. □ α = 0.01 □ α = 0.05 □ α = 0.10 □ none of the above scratch work (not graded) standard normal distribution step 1: select one - tailed or two - tailed. ⚪ one - tailed ⚪ two - tailed step 2: enter the critical value(s). (round to 3 decimal places.) step 3: enter the test statistic. (round to 3 decimal places.)

Explanation:

Part (a)

Step 1: Critical values for \(\alpha = 0.01\)

For a two - tailed test with \(\alpha=0.01\), the area in each tail is \(\frac{\alpha}{2}=\frac{0.01}{2} = 0.005\). We need to find the \(z\) - score such that \(P(Z < - z_{\alpha/2})=0.005\) and \(P(Z>z_{\alpha/2}) = 0.005\). Using the standard normal table or a calculator, \(z_{\alpha/2}=z_{0.005}\approx2.576\). So the critical values are \(- 2.576\) and \(2.576\).

Step 2: Critical values for \(\alpha = 0.05\)

For \(\alpha = 0.05\), the area in each tail is \(\frac{\alpha}{2}=\frac{0.05}{2}=0.025\). We find \(z_{\alpha/2}=z_{0.025}\approx1.960\). So the critical values are \(-1.960\) and \(1.960\).

Step 3: Critical values for \(\alpha=0.10\)

For \(\alpha = 0.10\), the area in each tail is \(\frac{\alpha}{2}=\frac{0.10}{2} = 0.05\). We find \(z_{\alpha/2}=z_{0.05}\approx1.645\). So the critical values are \(- 1.645\) and \(1.645\).

Part (b)

We have a test statistic \(z = 2.497\). We compare the absolute value of the test statistic \(|z|=2.497\) with the critical values for each \(\alpha\):

• For \(\alpha = 0.01\): The critical value is \(z_{\alpha/2}=2.576\). Since \(2.497<2.576\), we fail to reject the null hypothesis.
• For \(\alpha=0.05\): The critical value is \(z_{\alpha/2} = 1.960\). Since \(2.497>1.960\), we reject the null hypothesis.
• For \(\alpha = 0.10\): The critical value is \(z_{\alpha/2}=1.645\). Since \(2.497>1.645\), we reject the null hypothesis.

Final Answers

Part (a)

• Critical values at \(\alpha = 0.01\): \(-2.576\) and \(2.576\)
• Critical values at \(\alpha=0.05\): \(- 1.960\) and \(1.960\)
• Critical values at \(\alpha = 0.10\): \(-1.645\) and \(1.645\)

Part (b)

The levels of significance for which the null hypothesis is rejected are \(\alpha = 0.05\) and \(\alpha=0.10\)

So, for part (b), the answer is \(\alpha = 0.05\), \(\alpha = 0.10\)

Answer:

Part (a)

Step 1: Critical values for \(\alpha = 0.01\)

For a two - tailed test with \(\alpha=0.01\), the area in each tail is \(\frac{\alpha}{2}=\frac{0.01}{2} = 0.005\). We need to find the \(z\) - score such that \(P(Z < - z_{\alpha/2})=0.005\) and \(P(Z>z_{\alpha/2}) = 0.005\). Using the standard normal table or a calculator, \(z_{\alpha/2}=z_{0.005}\approx2.576\). So the critical values are \(- 2.576\) and \(2.576\).

Step 2: Critical values for \(\alpha = 0.05\)

For \(\alpha = 0.05\), the area in each tail is \(\frac{\alpha}{2}=\frac{0.05}{2}=0.025\). We find \(z_{\alpha/2}=z_{0.025}\approx1.960\). So the critical values are \(-1.960\) and \(1.960\).

Step 3: Critical values for \(\alpha=0.10\)

For \(\alpha = 0.10\), the area in each tail is \(\frac{\alpha}{2}=\frac{0.10}{2} = 0.05\). We find \(z_{\alpha/2}=z_{0.05}\approx1.645\). So the critical values are \(- 1.645\) and \(1.645\).

Part (b)

We have a test statistic \(z = 2.497\). We compare the absolute value of the test statistic \(|z|=2.497\) with the critical values for each \(\alpha\):

• For \(\alpha = 0.01\): The critical value is \(z_{\alpha/2}=2.576\). Since \(2.497<2.576\), we fail to reject the null hypothesis.
• For \(\alpha=0.05\): The critical value is \(z_{\alpha/2} = 1.960\). Since \(2.497>1.960\), we reject the null hypothesis.
• For \(\alpha = 0.10\): The critical value is \(z_{\alpha/2}=1.645\). Since \(2.497>1.645\), we reject the null hypothesis.

Final Answers

Part (a)

• Critical values at \(\alpha = 0.01\): \(-2.576\) and \(2.576\)
• Critical values at \(\alpha=0.05\): \(- 1.960\) and \(1.960\)
• Critical values at \(\alpha = 0.10\): \(-1.645\) and \(1.645\)

Part (b)

The levels of significance for which the null hypothesis is rejected are \(\alpha = 0.05\) and \(\alpha=0.10\)

So, for part (b), the answer is \(\alpha = 0.05\), \(\alpha = 0.10\)