Question
A test is made of Hv0 : u=18 versus Hv1: u does not = 18. A sample of size n = 45 is drawn, and x=16 . The population standard deviation is o=6.
(a) Compute the value of the test statistic z.
(b) Is Hv0 rejected at the a = 0.01 level?
(c) Is Hv0 rejected at the a= 0.05 level?
(a) Compute the value of the test statistic z.
(b) Is Hv0 rejected at the a = 0.01 level?
(c) Is Hv0 rejected at the a= 0.05 level?
Answers
Answer
Choose the correct type of hypothesis test. Then find the critical values for a=0.01 and a=0.05. Round your answers to three decimal places, if necessary.
Is Hv1 right-tailed, left-tailed or two-tailed? What are the critical values for a=0.01? the critical values for a=0.05?
Is Hv1 right-tailed, left-tailed or two-tailed? What are the critical values for a=0.01? the critical values for a=0.05?
Answered by
GPT-4o mini
To solve this hypothesis testing problem, we can follow these steps.
### (a) Compute the value of the test statistic z.
The formula for the z-test statistic when testing a population mean is given by:
\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]
where:
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis,
- \(\sigma\) is the population standard deviation, and
- \(n\) is the sample size.
Given:
- \(\bar{x} = 16\)
- \(\mu_0 = 18\)
- \(\sigma = 6\)
- \(n = 45\)
Plugging in the values:
\[
z = \frac{16 - 18}{\frac{6}{\sqrt{45}}}
\]
Calculating the denominator:
\[
\frac{6}{\sqrt{45}} \approx \frac{6}{6.7082} \approx 0.8944
\]
Now substituting:
\[
z = \frac{-2}{0.8944} \approx -2.2361
\]
Therefore,
\[
z \approx -2.24
\]
### (b) Is \(H_0\) rejected at the \(\alpha = 0.01\) level?
For a two-tailed test at \(\alpha = 0.01\), we need to find the critical z-values. The critical values are:
\[
\pm z_{\alpha/2} = \pm z_{0.005}
\]
From the standard normal distribution table, \(z_{0.005} \approx 2.576\).
Thus, our rejection regions are:
- \(z < -2.576\)
- \(z > 2.576\)
Since our calculated \(z \approx -2.24\), which does not fall into the rejection region (\(-2.576 < -2.24 < 2.576\)), we **do not reject** \(H_0\) at the \(\alpha = 0.01\) level.
### (c) Is \(H_0\) rejected at the \(\alpha = 0.05\) level?
For a two-tailed test at \(\alpha = 0.05\), the critical z-values are:
\[
\pm z_{\alpha/2} = \pm z_{0.025}
\]
From the standard normal distribution table, \(z_{0.025} \approx 1.96\).
Thus, our rejection regions are:
- \(z < -1.96\)
- \(z > 1.96\)
Since our calculated \(z \approx -2.24\), which falls into the rejection region (\(-2.24 < -1.96\)), we **reject** \(H_0\) at the \(\alpha = 0.05\) level.
### Summary:
(a) The value of the test statistic \(z \approx -2.24\).
(b) \(H_0\) is **not rejected** at \(\alpha = 0.01\).
(c) \(H_0\) is **rejected** at \(\alpha = 0.05\).
### (a) Compute the value of the test statistic z.
The formula for the z-test statistic when testing a population mean is given by:
\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]
where:
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis,
- \(\sigma\) is the population standard deviation, and
- \(n\) is the sample size.
Given:
- \(\bar{x} = 16\)
- \(\mu_0 = 18\)
- \(\sigma = 6\)
- \(n = 45\)
Plugging in the values:
\[
z = \frac{16 - 18}{\frac{6}{\sqrt{45}}}
\]
Calculating the denominator:
\[
\frac{6}{\sqrt{45}} \approx \frac{6}{6.7082} \approx 0.8944
\]
Now substituting:
\[
z = \frac{-2}{0.8944} \approx -2.2361
\]
Therefore,
\[
z \approx -2.24
\]
### (b) Is \(H_0\) rejected at the \(\alpha = 0.01\) level?
For a two-tailed test at \(\alpha = 0.01\), we need to find the critical z-values. The critical values are:
\[
\pm z_{\alpha/2} = \pm z_{0.005}
\]
From the standard normal distribution table, \(z_{0.005} \approx 2.576\).
Thus, our rejection regions are:
- \(z < -2.576\)
- \(z > 2.576\)
Since our calculated \(z \approx -2.24\), which does not fall into the rejection region (\(-2.576 < -2.24 < 2.576\)), we **do not reject** \(H_0\) at the \(\alpha = 0.01\) level.
### (c) Is \(H_0\) rejected at the \(\alpha = 0.05\) level?
For a two-tailed test at \(\alpha = 0.05\), the critical z-values are:
\[
\pm z_{\alpha/2} = \pm z_{0.025}
\]
From the standard normal distribution table, \(z_{0.025} \approx 1.96\).
Thus, our rejection regions are:
- \(z < -1.96\)
- \(z > 1.96\)
Since our calculated \(z \approx -2.24\), which falls into the rejection region (\(-2.24 < -1.96\)), we **reject** \(H_0\) at the \(\alpha = 0.05\) level.
### Summary:
(a) The value of the test statistic \(z \approx -2.24\).
(b) \(H_0\) is **not rejected** at \(\alpha = 0.01\).
(c) \(H_0\) is **rejected** at \(\alpha = 0.05\).
Answered by
GPT-4o mini
In your original hypothesis test, you stated the null hypothesis \(H_0: \mu = 18\) versus the alternative hypothesis \(H_1: \mu \neq 18\). Since the alternative hypothesis is testing for a mean not equal to a specified value, it indicates a two-tailed test.
### Type of Hypothesis Test:
- **Type**: Two-tailed test
### Critical Values:
For a two-tailed test, the critical values will be determined based on the specified significance levels, \(\alpha\).
1. **For \(\alpha = 0.01\)**:
- We divide the significance level by 2 for a two-tailed test:
\[
\alpha/2 = 0.005
\]
- Using the standard normal (z) distribution, the critical values at this level are:
\[
z_{0.005} \approx \pm 2.576
\]
- So, the critical values for \(\alpha = 0.01\) are:
\[
-2.576 \quad \text{and} \quad 2.576
\]
2. **For \(\alpha = 0.05\)**:
- Similarly, divide the significance level by 2:
\[
\alpha/2 = 0.025
\]
- From the standard normal distribution, the critical values at this level are:
\[
z_{0.025} \approx \pm 1.96
\]
- Therefore, the critical values for \(\alpha = 0.05\) are:
\[
-1.96 \quad \text{and} \quad 1.96
\]
### Summary:
- **Type of Test**: Two-tailed
- **Critical Values for \(\alpha = 0.01\)**: \(-2.576\) and \(2.576\)
- **Critical Values for \(\alpha = 0.05\)**: \(-1.96\) and \(1.96\)
### Type of Hypothesis Test:
- **Type**: Two-tailed test
### Critical Values:
For a two-tailed test, the critical values will be determined based on the specified significance levels, \(\alpha\).
1. **For \(\alpha = 0.01\)**:
- We divide the significance level by 2 for a two-tailed test:
\[
\alpha/2 = 0.005
\]
- Using the standard normal (z) distribution, the critical values at this level are:
\[
z_{0.005} \approx \pm 2.576
\]
- So, the critical values for \(\alpha = 0.01\) are:
\[
-2.576 \quad \text{and} \quad 2.576
\]
2. **For \(\alpha = 0.05\)**:
- Similarly, divide the significance level by 2:
\[
\alpha/2 = 0.025
\]
- From the standard normal distribution, the critical values at this level are:
\[
z_{0.025} \approx \pm 1.96
\]
- Therefore, the critical values for \(\alpha = 0.05\) are:
\[
-1.96 \quad \text{and} \quad 1.96
\]
### Summary:
- **Type of Test**: Two-tailed
- **Critical Values for \(\alpha = 0.01\)**: \(-2.576\) and \(2.576\)
- **Critical Values for \(\alpha = 0.05\)**: \(-1.96\) and \(1.96\)