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 = - 1.359. 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 a two - tailed hypothesis test with a standard normal distribution, the critical values are found by determining the z - scores that leave $\frac{\alpha}{2}$ area in each tail.

Step 1: Critical values for $\alpha = 0.01$

The area in each tail is $\frac{0.01}{2}=0.005$. We need to find the $z$ - score such that $P(Z < z)=0.005$ and $P(Z>z) = 0.005$. Looking up in the standard normal table (or using a calculator), the $z$ - score corresponding to a left - tail area of $0.005$ is $z=- 2.576$ and the $z$ - score corresponding to a right - tail area of $0.005$ (or a left - tail area of $1 - 0.005=0.995$) is $z = 2.576$.

Step 2: Critical values for $\alpha=0.05$

The area in each tail is $\frac{0.05}{2}=0.025$. The $z$ - score corresponding to a left - tail area of $0.025$ is $z=-1.960$ and the $z$ - score corresponding to a left - tail area of $1 - 0.025 = 0.975$ is $z = 1.960$.

Step 3: Critical values for $\alpha = 0.10$

The area in each tail is $\frac{0.10}{2}=0.05$. The $z$ - score corresponding to a left - tail area of $0.05$ is $z=-1.645$ and the $z$ - score corresponding to a left - tail area of $1 - 0.05=0.95$ is $z = 1.645$.

Part (b)

To determine if we reject the null hypothesis, we compare the test statistic $z=-1.359$ with the critical values for each $\alpha$.

• For $\alpha = 0.01$: The critical values are $\pm2.576$. Since $- 2.576<-1.359<2.576$, we fail to reject the null hypothesis.
• For $\alpha = 0.05$: The critical values are $\pm1.960$. Since $-1.960<-1.359<1.960$, we fail to reject the null hypothesis.
• For $\alpha=0.10$: The critical values are $\pm1.645$. Since $-1.645<-1.359<1.645$, we fail to reject the null hypothesis.

Part (a) Answers:

• 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:

None of the above (because the test statistic $z = - 1.359$ lies within the critical value ranges for $\alpha=0.01$, $\alpha = 0.05$, and $\alpha=0.10$)

Answer:

To determine if we reject the null hypothesis, we compare the test statistic $z=-1.359$ with the critical values for each $\alpha$.

• For $\alpha = 0.01$: The critical values are $\pm2.576$. Since $- 2.576<-1.359<2.576$, we fail to reject the null hypothesis.
• For $\alpha = 0.05$: The critical values are $\pm1.960$. Since $-1.960<-1.359<1.960$, we fail to reject the null hypothesis.
• For $\alpha=0.10$: The critical values are $\pm1.645$. Since $-1.645<-1.359<1.645$, we fail to reject the null hypothesis.

Part (a) Answers:

• 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:

None of the above (because the test statistic $z = - 1.359$ lies within the critical value ranges for $\alpha=0.01$, $\alpha = 0.05$, and $\alpha=0.10$)