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 = 2.509. 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)

For \(\alpha = 0.01\):

Step1: Determine the tail area

For a two - tailed test, the area in each tail is \(\frac{\alpha}{2}=\frac{0.01}{2} = 0.005\). The area to the left of the lower critical value is \(0.005\), and the area to the left of the upper critical value is \(1 - 0.005=0.995\).

Step2: Find the critical values

Using the standard normal distribution table (or z - table), the \(z\) - value corresponding to an area of \(0.005\) (lower tail) is \(z=- 2.576\) (since \(P(Z < - 2.576)=0.005\)) and the \(z\) - value corresponding to an area of \(0.995\) (upper tail) is \(z = 2.576\) (since \(P(Z < 2.576)=0.995\)).

For \(\alpha=0.05\):

Step1: Determine the tail area

For a two - tailed test, the area in each tail is \(\frac{\alpha}{2}=\frac{0.05}{2}=0.025\). The area to the left of the lower critical value is \(0.025\), and the area to the left of the upper critical value is \(1 - 0.025 = 0.975\).

Step2: Find the critical values

Using the standard normal distribution table, the \(z\) - value corresponding to an area of \(0.025\) (lower tail) is \(z=-1.960\) (since \(P(Z < - 1.960)=0.025\)) and the \(z\) - value corresponding to an area of \(0.975\) (upper tail) is \(z = 1.960\) (since \(P(Z < 1.960)=0.975\)).

For \(\alpha = 0.10\):

Step1: Determine the tail area

For a two - tailed test, the area in each tail is \(\frac{\alpha}{2}=\frac{0.10}{2}=0.05\). The area to the left of the lower critical value is \(0.05\), and the area to the left of the upper critical value is \(1 - 0.05=0.95\).

Step2: Find the critical values

Using the standard normal distribution table, the \(z\) - value corresponding to an area of \(0.05\) (lower tail) is \(z=-1.645\) (since \(P(Z < - 1.645)=0.05\)) and the \(z\) - value corresponding to an area of \(0.95\) (upper tail) is \(z = 1.645\) (since \(P(Z < 1.645)=0.95\)).

Part (b)

We have the test statistic \(z = 2.509\). We compare the absolute value of the test statistic \(|z|=2.509\) with the absolute values of the critical values for each \(\alpha\):

• For \(\alpha = 0.01\): The critical values are \(\pm2.576\). Since \(|2.509|=2.509<2.576\), we fail to reject the null hypothesis.
• For \(\alpha=0.05\): The critical values are \(\pm1.960\). Since \(|2.509| = 2.509>1.960\), we reject the null hypothesis.
• For \(\alpha = 0.10\): The critical values are \(\pm1.645\). Since \(|2.509|=2.509 > 1.645\), 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 would be rejected are \(\alpha = 0.05\) and \(\alpha=0.10\)

Part (a) Answer:

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

Part (b) Answer:

\(\alpha = 0.05\), \(\alpha=0.10\)

Answer:

Part (a)

For \(\alpha = 0.01\):

Step1: Determine the tail area

For a two - tailed test, the area in each tail is \(\frac{\alpha}{2}=\frac{0.01}{2} = 0.005\). The area to the left of the lower critical value is \(0.005\), and the area to the left of the upper critical value is \(1 - 0.005=0.995\).

Step2: Find the critical values

Using the standard normal distribution table (or z - table), the \(z\) - value corresponding to an area of \(0.005\) (lower tail) is \(z=- 2.576\) (since \(P(Z < - 2.576)=0.005\)) and the \(z\) - value corresponding to an area of \(0.995\) (upper tail) is \(z = 2.576\) (since \(P(Z < 2.576)=0.995\)).

For \(\alpha=0.05\):

Step1: Determine the tail area

For a two - tailed test, the area in each tail is \(\frac{\alpha}{2}=\frac{0.05}{2}=0.025\). The area to the left of the lower critical value is \(0.025\), and the area to the left of the upper critical value is \(1 - 0.025 = 0.975\).

Step2: Find the critical values

Using the standard normal distribution table, the \(z\) - value corresponding to an area of \(0.025\) (lower tail) is \(z=-1.960\) (since \(P(Z < - 1.960)=0.025\)) and the \(z\) - value corresponding to an area of \(0.975\) (upper tail) is \(z = 1.960\) (since \(P(Z < 1.960)=0.975\)).

For \(\alpha = 0.10\):

Step1: Determine the tail area

For a two - tailed test, the area in each tail is \(\frac{\alpha}{2}=\frac{0.10}{2}=0.05\). The area to the left of the lower critical value is \(0.05\), and the area to the left of the upper critical value is \(1 - 0.05=0.95\).

Step2: Find the critical values

Using the standard normal distribution table, the \(z\) - value corresponding to an area of \(0.05\) (lower tail) is \(z=-1.645\) (since \(P(Z < - 1.645)=0.05\)) and the \(z\) - value corresponding to an area of \(0.95\) (upper tail) is \(z = 1.645\) (since \(P(Z < 1.645)=0.95\)).

Part (b)

We have the test statistic \(z = 2.509\). We compare the absolute value of the test statistic \(|z|=2.509\) with the absolute values of the critical values for each \(\alpha\):

• For \(\alpha = 0.01\): The critical values are \(\pm2.576\). Since \(|2.509|=2.509<2.576\), we fail to reject the null hypothesis.
• For \(\alpha=0.05\): The critical values are \(\pm1.960\). Since \(|2.509| = 2.509>1.960\), we reject the null hypothesis.
• For \(\alpha = 0.10\): The critical values are \(\pm1.645\). Since \(|2.509|=2.509 > 1.645\), 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 would be rejected are \(\alpha = 0.05\) and \(\alpha=0.10\)

Part (a) Answer:

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

Part (b) Answer:

\(\alpha = 0.05\), \(\alpha=0.10\)