7.1-11 ### Statistical Analysis Problems#### Problem on Estimation and Confidence IntervalsGiven the following observations of the random variable $ X $, assumed to be distributed as $ N(\mu, \sigma^2) $:- Observations: 3.1, 3.3, 4.5, 2.8, 3.5, 3.5, 3.7, 4.2, 3.9, 3.3Use these observations to:1. **Find a point estimate of $ \mu $.**2. **Find a point estimate of $ \sigma $.**3. **Find a 95% one-sided confidence interval for $ \mu $ that provides an upper bound for $ \mu $.**#### Problem on Confidence Intervals in Marathon Data**7.1-11**: The heart rates (in beats per minute) of 41 randomly selected finishers of the Chicago Marathon, five minutes after they completed the race, had a sample mean $ \bar{x} = 132 $ and a sample variance $ s^2 = 105 $. Assuming that the heart rates of all finishers are normally distributed, obtain a 95% confidence interval for their mean.---#### Problem on Calibration in Nuclear Physics**7.1-12**: In nuclear physics, detectors are often used to measure the energy of a particle. To calibrate a detector, particles of known energy are directed into it. The values of signals from 15 different detectors, for the same energy, are:- Values: 260, 216, 259, 206, 265, 284, 291, 229, 232, 250, 225, 242, 240, 252, 236**Task:**1. **Find a 95% confidence interval for these observed values.**

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Description: 7.1-11 ### Statistical Analysis Problems#### Problem on Estimation and Confidence IntervalsGiven the following observations of the random variable $ X $, assumed to be distributed as $ N(\mu, \sigma^2) $:- Observations: 3.1, 3.3, 4.5, 2.8, 3.5, 3

Given that, The degree of freedom is, df=n-1 =41-1 =40 Critical value: By using the t-tables,…

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provides an up 7.1-11. The heart rates (in beats per minute) of 41 ran- domly selected finishers of the Chicago Marathon, five minutes after they completed the race, had sample mean 132 and sample variance s? the heart rates of all finishers of the Chicago Marathon five minutes after completing the race are normally dis- tributed, obtain a 95% confidence interval for their mean 105. Assuming that %3D %3D

provides an up 7.1-11. The heart rates (in beats per minute) of 41 ran- domly selected finishers of the Chicago Marathon, five minutes after they completed the race, had sample mean 132 and sample variance s? the heart rates of all finishers of the Chicago Marathon five minutes after completing the race are normally dis- tributed, obtain a 95% confidence interval for their mean 105. Assuming that %3D %3D

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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Question

7.1-11

### Statistical Analysis Problems

#### Problem on Estimation and Confidence Intervals

Given the following observations of the random variable $ X $, assumed to be distributed as $ N(\mu, \sigma^2) $:

- Observations: 3.1, 3.3, 4.5, 2.8, 3.5, 3.5, 3.7, 4.2, 3.9, 3.3

Use these observations to:
1. **Find a point estimate of $ \mu $.**
2. **Find a point estimate of $ \sigma $.**
3. **Find a 95% one-sided confidence interval for $ \mu $ that provides an upper bound for $ \mu $.**

#### Problem on Confidence Intervals in Marathon Data

**7.1-11**: The heart rates (in beats per minute) of 41 randomly selected finishers of the Chicago Marathon, five minutes after they completed the race, had a sample mean $ \bar{x} = 132 $ and a sample variance $ s^2 = 105 $. Assuming that the heart rates of all finishers are normally distributed, obtain a 95% confidence interval for their mean.

---

#### Problem on Calibration in Nuclear Physics

**7.1-12**: In nuclear physics, detectors are often used to measure the energy of a particle. To calibrate a detector, particles of known energy are directed into it. The values of signals from 15 different detectors, for the same energy, are:

- Values: 260, 216, 259, 206, 265, 284, 291, 229, 232, 250, 225, 242, 240, 252, 236

**Task:**
1. **Find a 95% confidence interval for these observed values.**

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