Question

a researcher is going to perform a right - 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 right - tailed test. for each potential choice for the level of significance, find the critical value. round your answers to three decimal places. critical value at α = 0.01: critical value at α = 0.05: critical value at α = 0.10: (b) after choosing the level of significance, the researcher takes a sample and finds that the value of the test statistic is z = 1.544. for which of the levels of significance would the null hypothesis not 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)

Step1: For \(\alpha = 0.01\) (right - tailed)

We need to find the \(z\) - value such that \(P(Z>z)=\alpha = 0.01\), which is equivalent to \(P(Z\leq z)=1 - 0.01=0.99\). Looking up the value in the standard normal distribution table (or using a calculator with the inverse - normal function), the \(z\) - value corresponding to a cumulative probability of \(0.99\) is approximately \(2.326\).

Step2: For \(\alpha = 0.05\) (right - tailed)

We need to find the \(z\) - value such that \(P(Z > z)=0.05\), which is equivalent to \(P(Z\leq z)=1 - 0.05 = 0.95\). Looking up the value in the standard normal distribution table (or using a calculator with the inverse - normal function), the \(z\) - value corresponding to a cumulative probability of \(0.95\) is approximately \(1.645\).

Step3: For \(\alpha = 0.10\) (right - tailed)

We need to find the \(z\) - value such that \(P(Z>z)=0.10\), which is equivalent to \(P(Z\leq z)=1 - 0.10 = 0.90\). Looking up the value in the standard normal distribution table (or using a calculator with the inverse - normal function), the \(z\) - value corresponding to a cumulative probability of \(0.90\) is approximately \(1.282\).

We know that in a right - tailed hypothesis test, we reject the null hypothesis if the test statistic \(z\) (which is \(1.544\) in this case) is greater than the critical value. We do not reject the null hypothesis if \(z\leq\) critical value.

• For \(\alpha = 0.01\), critical value \(z_{0.01}=2.326\). Since \(1.544<2.326\), we do not reject the null hypothesis.
• For \(\alpha = 0.05\), critical value \(z_{0.05}=1.645\). Since \(1.544<1.645\), we do not reject the null hypothesis.
• For \(\alpha = 0.10\), critical value \(z_{0.10}=1.282\). Since \(1.544 > 1.282\), we reject the null hypothesis.

To determine when the null hypothesis is not rejected, we compare the test statistic \(z = 1.544\) with the critical values from part (a). For \(\alpha=0.01\), critical value is \(2.326\) (\(1.544<2.326\), do not reject). For \(\alpha = 0.05\), critical value is \(1.645\) (\(1.544<1.645\), do not reject). For \(\alpha=0.10\), critical value is \(1.282\) (\(1.544 > 1.282\), reject).

Answer:

s for part (a):
Critical value at \(\alpha = 0.01\): \(2.326\)
Critical value at \(\alpha = 0.05\): \(1.645\)
Critical value at \(\alpha = 0.10\): \(1.282\)

Part (b)