Question
What is the sample size required to obtain a test statistic of -1.237 for the given hypothesis test, assuming the same significance level and null hypothesis?
Answers
GPT-4o mini
To determine the required sample size to obtain a specific test statistic, we need to know a few key details about the hypothesis test being conducted. The test statistic is typically calculated using the formula:
\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]
Where:
- \(z\) is the test statistic (in this case, -1.237),
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
To determine the sample size \(n\), we need to rearrange the formula:
\[
n = \left( \frac{\sigma}{\bar{x} - \mu_0} \right)^2 \cdot z^2
\]
Without specific values for \(\bar{x}\), \(\mu_0\), and \(\sigma\), we cannot compute \(n\) directly. However, here's how you would go about finding the sample size if you had the specific values:
1. **Identify or determine the population standard deviation (\(\sigma\))**.
2. **Specify the values of the sample mean (\(\bar{x}\)) and null hypothesis mean (\(\mu_0\))**.
3. **Use the known test statistic (z = -1.237)** to plug it into the rearranged formula.
Do you have the values for \(\sigma\), \(\bar{x}\), and \(\mu_0\)? If you provide those, I can help you calculate the required sample size.
\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]
Where:
- \(z\) is the test statistic (in this case, -1.237),
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
To determine the sample size \(n\), we need to rearrange the formula:
\[
n = \left( \frac{\sigma}{\bar{x} - \mu_0} \right)^2 \cdot z^2
\]
Without specific values for \(\bar{x}\), \(\mu_0\), and \(\sigma\), we cannot compute \(n\) directly. However, here's how you would go about finding the sample size if you had the specific values:
1. **Identify or determine the population standard deviation (\(\sigma\))**.
2. **Specify the values of the sample mean (\(\bar{x}\)) and null hypothesis mean (\(\mu_0\))**.
3. **Use the known test statistic (z = -1.237)** to plug it into the rearranged formula.
Do you have the values for \(\sigma\), \(\bar{x}\), and \(\mu_0\)? If you provide those, I can help you calculate the required sample size.