A researcher is interested in finding a 90% confidence interval for the mean number minutes students are concentrating on their professor during a one hour statistics lecture. The study included 114 students who averaged 32.2 minutes concentrating on their professor during the hour lecture. The standard deviation was 11.2 minutes. Round answers to 3 decimal places where possible. a. To compute the confidence interval use a z distribution. b. With 90% confidence the population mean minutes of concentration is between and minutes. c. If many groups of 114 randomly selected members are studied, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population mean minutes of concentration and about percent will not contain the true population mean minutes of concentration.
A researcher is interested in finding a 90% confidence interval for the mean number minutes students are concentrating on their professor during a one hour statistics lecture. The study included 114 students who averaged 32.2 minutes concentrating on their professor during the hour lecture. The standard deviation was 11.2 minutes. Round answers to 3 decimal places where possible. a. To compute the confidence interval use a z distribution. b. With 90% confidence the population mean minutes of concentration is between and minutes. c. If many groups of 114 randomly selected members are studied, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population mean minutes of concentration and about percent will not contain the true population mean minutes of concentration.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![### Confidence Intervals in Statistics
**Research Scenario:**
A researcher is interested in finding a 90% confidence interval for the mean number of minutes students are concentrating on their professor during a one-hour statistics lecture. The study included 114 students who averaged 32.2 minutes concentrating on their professor during the hour lecture. The standard deviation was 11.2 minutes. Round answers to 3 decimal places where possible.
---
**Questions:**
**a. To compute the confidence interval use a \( z \) distribution.**
**b. With 90% confidence the population mean minutes of concentration is between** \( \_\_\_ \) **and** \( \_\_\_ \) **minutes.**
**c. If many groups of 114 randomly selected members are studied, then a different confidence interval would be produced from each group. About** \( \_\_\_ \) **percent of these confidence intervals will contain the true population mean minutes of concentration and about** \( \_\_\_ \) **percent will not contain the true population mean minutes of concentration.**
---
**Details:**
1. **a.** For constructing a confidence interval using the \( z \) distribution, you should recognize that you will be using the standard normal distribution table to find the critical value associated with a 90% confidence level.
2. **b.** To find the bounds of the confidence interval, you will calculate the margin of error using the critical \( z \)-value, the standard deviation, and the sample size. The formula for the confidence interval is:
\[ \text{CI} = \bar{x} \pm (z^* \cdot \frac{s}{\sqrt{n}}) \]
where:
- \( \bar{x} \) is the sample mean,
- \( z^* \) is the critical \( z \)-value,
- \( s \) is the standard deviation,
- \( n \) is the sample size.
3. **c.** The concept of confidence intervals relies on repeated sampling. If we repeatedly draw samples of 114 students from the population and compute the confidence interval for each sample, we can expect about 90% of these intervals to contain the true population mean, while about 10% will not.
This section would help students understand how to construct and interpret confidence intervals, especially in the context of statistical data involving student concentration during lectures.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1a73d04-4829-40e4-8e78-0a30aa84e71e%2F98dfe563-36e4-4bb5-8183-66c08ff2f7e4%2Frkogii_processed.png&w=3840&q=75)
Transcribed Image Text:### Confidence Intervals in Statistics
**Research Scenario:**
A researcher is interested in finding a 90% confidence interval for the mean number of minutes students are concentrating on their professor during a one-hour statistics lecture. The study included 114 students who averaged 32.2 minutes concentrating on their professor during the hour lecture. The standard deviation was 11.2 minutes. Round answers to 3 decimal places where possible.
---
**Questions:**
**a. To compute the confidence interval use a \( z \) distribution.**
**b. With 90% confidence the population mean minutes of concentration is between** \( \_\_\_ \) **and** \( \_\_\_ \) **minutes.**
**c. If many groups of 114 randomly selected members are studied, then a different confidence interval would be produced from each group. About** \( \_\_\_ \) **percent of these confidence intervals will contain the true population mean minutes of concentration and about** \( \_\_\_ \) **percent will not contain the true population mean minutes of concentration.**
---
**Details:**
1. **a.** For constructing a confidence interval using the \( z \) distribution, you should recognize that you will be using the standard normal distribution table to find the critical value associated with a 90% confidence level.
2. **b.** To find the bounds of the confidence interval, you will calculate the margin of error using the critical \( z \)-value, the standard deviation, and the sample size. The formula for the confidence interval is:
\[ \text{CI} = \bar{x} \pm (z^* \cdot \frac{s}{\sqrt{n}}) \]
where:
- \( \bar{x} \) is the sample mean,
- \( z^* \) is the critical \( z \)-value,
- \( s \) is the standard deviation,
- \( n \) is the sample size.
3. **c.** The concept of confidence intervals relies on repeated sampling. If we repeatedly draw samples of 114 students from the population and compute the confidence interval for each sample, we can expect about 90% of these intervals to contain the true population mean, while about 10% will not.
This section would help students understand how to construct and interpret confidence intervals, especially in the context of statistical data involving student concentration during lectures.
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