A Simple Random Sample Is Drawn From A Normally Distributed Population, And The Mean Is Found To Be 138.Given The Confidence Levels And Their Corresponding $z^*$-scores:$[ \begin{tabular}{|c|c|c|c|} \hline \text{Confidence Level} & 90%
Introduction
In statistics, a simple random sample is a subset of individuals selected from a larger population, where every individual has an equal chance of being selected. This type of sampling is used to make inferences about the population based on the sample. In this article, we will discuss the concept of confidence levels and their corresponding z*-scores, using a simple random sample drawn from a normally distributed population.
Confidence Level and its Importance
The confidence level is a measure of the reliability of a statistical estimate. It represents the probability that the true population parameter lies within a certain range of the sample estimate. In other words, it is the probability that the sample mean is close to the true population mean. The confidence level is usually expressed as a percentage, and it is denoted by the symbol α (alpha).
For example, if we have a 90% confidence level, it means that we are 90% confident that the true population mean lies within the range of the sample mean plus or minus 1.645 standard errors. The confidence level is an important concept in statistics because it helps us to make inferences about the population based on the sample.
z-scores and their Corresponding Confidence Levels*
The z*-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the formula:
z* = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation.
The z*-score is used to determine the confidence level of a statistical estimate. The corresponding z*-scores for different confidence levels are:
| Confidence Level | z*-score |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
Understanding the z-scores*
The z*-scores are used to determine the confidence level of a statistical estimate. The z*-score represents the number of standard deviations an observation is away from the mean. For example, if we have a z*-score of 1.645, it means that the observation is 1.645 standard deviations away from the mean.
The z*-scores are used to determine the confidence level of a statistical estimate by comparing it to the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The z*-scores are used to determine the probability that the true population parameter lies within a certain range of the sample estimate.
Example: A Simple Random Sample Drawn from a Normally Distributed Population
Let's consider an example where a simple random sample is drawn from a normally distributed population with a mean of 138. The sample size is 100, and the standard deviation is 10.
| Sample | Mean | Standard Deviation |
|---|---|---|
| 1 | 140 | 12 |
| 2 | 135 | 11 |
| 3 | 142 | 13 |
| ... | ... | ... |
| 100 | 138 | 10 |
The sample mean is 138, and the sample standard deviation is 10. We want to determine the confidence level of the sample mean.
Calculating the Confidence Interval
To calculate the confidence interval, we need to calculate the standard error of the mean. The standard error of the mean is calculated using the formula:
SE = σ / √n
where σ is the standard deviation, and n is the sample size.
SE = 10 / √100 SE = 1
The confidence interval is calculated using the formula:
CI = X̄ ± (z* × SE)
where XÌ„ is the sample mean, z* is the z*-score, and SE is the standard error of the mean.
CI = 138 ± (1.645 × 1) CI = 138 ± 1.645 CI = (136.355, 139.645)
Conclusion
In this article, we discussed the concept of confidence levels and their corresponding z*-scores, using a simple random sample drawn from a normally distributed population. We calculated the confidence interval of the sample mean and determined the confidence level of the sample estimate. The confidence level is an important concept in statistics because it helps us to make inferences about the population based on the sample.
References
- [1] Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Larson, R. E., & Farber, B. A. (2013). Elementary statistics: Picturing the world. Cengage Learning.
- [3] Agresti, A., & Franklin, C. (2014). Statistics: The art and science of learning from data. Pearson Education.
Discussion
The confidence level is a measure of the reliability of a statistical estimate. It represents the probability that the true population parameter lies within a certain range of the sample estimate. The confidence level is usually expressed as a percentage, and it is denoted by the symbol α (alpha).
The z*-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the formula:
z* = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation.
The z*-scores are used to determine the confidence level of a statistical estimate. The corresponding z*-scores for different confidence levels are:
| Confidence Level | z*-score |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
The confidence interval is calculated using the formula:
CI = X̄ ± (z* × SE)
where XÌ„ is the sample mean, z* is the z*-score, and SE is the standard error of the mean.
The confidence level is an important concept in statistics because it helps us to make inferences about the population based on the sample.
Questions
- What is the confidence level of a statistical estimate?
- What is the z*-score, and how is it calculated?
- What is the confidence interval, and how is it calculated?
- What is the standard error of the mean, and how is it calculated?
- What is the relationship between the confidence level and the z*-score?
Answers
- The confidence level is a measure of the reliability of a statistical estimate. It represents the probability that the true population parameter lies within a certain range of the sample estimate.
- The z*-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the formula:
z* = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation. 3. The confidence interval is calculated using the formula:
CI = X̄ ± (z* × SE)
where XÌ„ is the sample mean, z* is the z*-score, and SE is the standard error of the mean. 4. The standard error of the mean is calculated using the formula:
SE = σ / √n
Q1: What is the confidence level of a statistical estimate?
A1: The confidence level is a measure of the reliability of a statistical estimate. It represents the probability that the true population parameter lies within a certain range of the sample estimate. The confidence level is usually expressed as a percentage, and it is denoted by the symbol α (alpha).
Q2: What is the z-score, and how is it calculated?*
A2: The z*-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the formula:
z* = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation.
Q3: What is the confidence interval, and how is it calculated?
A3: The confidence interval is calculated using the formula:
CI = X̄ ± (z* × SE)
where XÌ„ is the sample mean, z* is the z*-score, and SE is the standard error of the mean.
Q4: What is the standard error of the mean, and how is it calculated?
A4: The standard error of the mean is calculated using the formula:
SE = σ / √n
where σ is the standard deviation, and n is the sample size.
Q5: What is the relationship between the confidence level and the z-score?*
A5: The confidence level and the z*-score are related in that the z*-score is used to determine the confidence level of a statistical estimate. The corresponding z*-scores for different confidence levels are:
| Confidence Level | z*-score |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
Q6: How do I determine the confidence level of a statistical estimate?
A6: To determine the confidence level of a statistical estimate, you need to calculate the z*-score using the formula:
z* = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation.
Then, you can use the corresponding z*-scores for different confidence levels to determine the confidence level of the statistical estimate.
Q7: What is the difference between a confidence interval and a prediction interval?
A7: A confidence interval is used to estimate the population parameter, while a prediction interval is used to predict a future observation. The confidence interval is calculated using the formula:
CI = X̄ ± (z* × SE)
where XÌ„ is the sample mean, z* is the z*-score, and SE is the standard error of the mean.
The prediction interval is calculated using the formula:
PI = X̄ ± (z* × SE) + (z* × σ)
where X̄ is the sample mean, z* is the z*-score, SE is the standard error of the mean, and σ is the standard deviation.
Q8: How do I choose the correct z-score for my statistical analysis?*
A8: To choose the correct z*-score for your statistical analysis, you need to determine the confidence level of your statistical estimate. The corresponding z*-scores for different confidence levels are:
| Confidence Level | z*-score |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
Choose the z*-score that corresponds to the desired confidence level.
Q9: What is the significance of the z-score in statistical analysis?*
A9: The z*-score is a measure of how many standard deviations an observation is away from the mean. It is used to determine the confidence level of a statistical estimate. The z*-score is an important concept in statistical analysis because it helps to determine the reliability of a statistical estimate.
Q10: How do I interpret the results of a statistical analysis using z-scores?*
A10: To interpret the results of a statistical analysis using z*-scores, you need to understand the meaning of the z*-score. The z*-score represents the number of standard deviations an observation is away from the mean. You can use the z*-score to determine the confidence level of a statistical estimate and to make inferences about the population.
Conclusion
In this Q&A article, we discussed the concept of confidence levels and z*-scores in statistical analysis. We answered questions about the confidence level, z*-score, confidence interval, standard error of the mean, and the relationship between the confidence level and the z*-score. We also discussed the significance of the z*-score in statistical analysis and how to interpret the results of a statistical analysis using z*-scores.