Suppose x has a normal distribution with s = 1.3. A random sample of size 51 has a sample mean of x = 20. **Confidence Interval Calculation Example**Suppose \( x \) has a normal distribution with \( s = 1.3 \). A random sample of size 51 has a sample mean of \( \overline{x} = 20 \).### (a) What conditions are necessary for your calculations? (Select all that apply)- [ ] uniform distribution of \( x \)- [x] normal distribution of \( x \)- [ ] \( \sigma \) is known- [x] \( \sigma \) is unknown - \checkmark### (b) Find a 95% confidence interval for \( \mu \). What is the margin of error? (Round your answers to two decimal places.)- **lower limit:** [ ]- **upper limit:** [ ]- **margin of error:** [ ]### (c) Interpret your results in the context of this problem.- [ ] The probability that this interval contains the true average is 0.05.- [ ] The probability that this interval contains the true average is 0.95.- [x] There is a 95% chance that the interval is one of the intervals containing the true average. - \checkmark- [ ] There is a 5% chance that the interval is one of the intervals containing the true average.In the setup given, the correct interpretation of a 95% confidence interval means there is a 95% chance that the interval calculated from the sample data includes the true population mean \( \mu \). This does not mean that there's a 95% probability that any specific interval calculated contains the true mean since the interval either does or does not contain the population parameter. Rather, if we were to take many samples and build a confidence interval from each sample, approximately 95% of those intervals would contain the true mean.

Given Suppose x has a normal distribution with s = 1.3. Sample of size (n) = 51 x = 20

Answered: ) What conditions are necessary for…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose x has a normal distribution with s = 1.3. A random sample of size 51 has a sample mean of x = 20.

**Confidence Interval Calculation Example**

Suppose \( x \) has a normal distribution with \( s = 1.3 \). A random sample of size 51 has a sample mean of \( \overline{x} = 20 \).

### (a) What conditions are necessary for your calculations? (Select all that apply)
- [ ] uniform distribution of \( x \)
- [x] normal distribution of \( x \)
- [ ] \( \sigma \) is known
- [x] \( \sigma \) is unknown
- \checkmark

### (b) Find a 95% confidence interval for \( \mu \). What is the margin of error? (Round your answers to two decimal places.)
- **lower limit:** [ ]
- **upper limit:** [ ]
- **margin of error:** [ ]

### (c) Interpret your results in the context of this problem.
- [ ] The probability that this interval contains the true average is 0.05.
- [ ] The probability that this interval contains the true average is 0.95.
- [x] There is a 95% chance that the interval is one of the intervals containing the true average.
- \checkmark
- [ ] There is a 5% chance that the interval is one of the intervals containing the true average.

In the setup given, the correct interpretation of a 95% confidence interval means there is a 95% chance that the interval calculated from the sample data includes the true population mean \( \mu \). This does not mean that there's a 95% probability that any specific interval calculated contains the true mean since the interval either does or does not contain the population parameter. Rather, if we were to take many samples and build a confidence interval from each sample, approximately 95% of those intervals would contain the true mean.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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