symmetric distribution. A random sample of size 16 has sample mean 10 and sample standard deviation 2. (a) Check Requirements Is it appropriate to use a Student's i distribution to compute a confidence interval for the population mean u? Explain. (b) Find a 90% confidence interval for μ. (c) Interpretation Explain the meaning of the confidence interval you computed.
symmetric distribution. A random sample of size 16 has sample mean 10 and sample standard deviation 2. (a) Check Requirements Is it appropriate to use a Student's i distribution to compute a confidence interval for the population mean u? Explain. (b) Find a 90% confidence interval for μ. (c) Interpretation Explain the meaning of the confidence interval you computed.
MATLAB: An Introduction with Applications
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Please answer number 11. All steps! Make sure to show work! Thanks
![### Basic Computation: Confidence Interval
Suppose \( x \) has a mound-shaped symmetric distribution. A random sample of size 16 has a sample mean of 10 and a sample standard deviation of 2.
1. **Check Requirements**
- **Question**: Is it appropriate to use a Student’s t distribution to compute a confidence interval for the population mean \( \mu \)? Explain.
- **Answer**: Yes, it is appropriate to use a Student’s t distribution because the sample size is small (n < 30) and the population is assumed to have a symmetric distribution. The Student’s t distribution is used instead of the normal distribution under these conditions.
2. **Find a 90% confidence interval for \( \mu \)**:
- To find a 90% confidence interval for the population mean \( \mu \), use the formula:
\[
\bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)
\]
Where:
- \( \bar{x} \) is the sample mean (10),
- \( t_{\alpha/2} \) is the t-value for a 90% confidence level with \( n-1 \) degrees of freedom (df = 15),
- \( s \) is the sample standard deviation (2),
- \( n \) is the sample size (16).
3. **Interpretation**
- **Question**: Explain the meaning of the confidence interval you computed.
- **Answer**: The 90% confidence interval provides a range of values that, with 90% confidence, contains the true population mean \( \mu \). This means that if we were to take many random samples of size 16 and construct a confidence interval from each sample, approximately 90% of those intervals would contain the true population mean.
By using these steps, you can understand how to determine and interpret a 90% confidence interval for a population mean using a sample's data.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc26c9a32-e967-4c64-99e6-f3828f7f80b4%2F1cd814d6-c573-43e6-adcb-464b4f52e531%2Fqux52he_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Basic Computation: Confidence Interval
Suppose \( x \) has a mound-shaped symmetric distribution. A random sample of size 16 has a sample mean of 10 and a sample standard deviation of 2.
1. **Check Requirements**
- **Question**: Is it appropriate to use a Student’s t distribution to compute a confidence interval for the population mean \( \mu \)? Explain.
- **Answer**: Yes, it is appropriate to use a Student’s t distribution because the sample size is small (n < 30) and the population is assumed to have a symmetric distribution. The Student’s t distribution is used instead of the normal distribution under these conditions.
2. **Find a 90% confidence interval for \( \mu \)**:
- To find a 90% confidence interval for the population mean \( \mu \), use the formula:
\[
\bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)
\]
Where:
- \( \bar{x} \) is the sample mean (10),
- \( t_{\alpha/2} \) is the t-value for a 90% confidence level with \( n-1 \) degrees of freedom (df = 15),
- \( s \) is the sample standard deviation (2),
- \( n \) is the sample size (16).
3. **Interpretation**
- **Question**: Explain the meaning of the confidence interval you computed.
- **Answer**: The 90% confidence interval provides a range of values that, with 90% confidence, contains the true population mean \( \mu \). This means that if we were to take many random samples of size 16 and construct a confidence interval from each sample, approximately 90% of those intervals would contain the true population mean.
By using these steps, you can understand how to determine and interpret a 90% confidence interval for a population mean using a sample's data.
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