12. confidence interval estimatibon.pptx

confidence interval

1. Confidence Interval

2. Significance level: In a hypothesis test, the significance level, alpha, is the probability of making the wrong decision when the null hypothesis is true. In statistical speak, another way of saying this is that it’s your probability of making a Type I error. Confidence level: The probability that if a poll/test/survey were repeated over and over again, the results obtained would be the same. A confidence level = 1 – alpha. Confidence interval: A range of results from a poll, experiment, or survey that would be expected to contain the population parameter of interest. For example, an average response. Confidence intervals are constructed using significance levels / confidence levels.

3. Factors that affect the width of a confidence interval include T he sample size, the variability in the sample, and the confidence level.  A larger sample produces a narrower confidence interval, while greater variability in the sample produces a wider confidence interval. 

4. There are basically two types of hypotheses and errors during data interpretation: Null Hypothesis Alternative Hypothesis The null hypothesis states no relationship between the variables, whereas the alternative hypothesis states a significant relationship between the variables in a data set. There are two types of errors associated with hypothesis testing. The two types of errors are related to drawing an incorrect conclusion. Type I error, generally known as false positives, rejects a null hypothesis that is true. On the other hand, Type II error, generally known as false negative, fails to reject a null hypothesis that is false . Brief about Hypothesis

5. Confidence interval for the mean with known variance When the population variance is known , we use the normal (Z) distribution to calculate the confidence interval for the mean. This situation is different from when the variance is unknown because the population variance ( ) is used directly, making the calculation slightly simpler.  

6. Confidence interval for the mean with known variance

7. Q. What is the 95% confidence interval for a sample mean of 100, given σ = 15, and a sample size of 36? Q: Why do we use the Z-distribution when σ is known? A: When the population standard deviation is known, the sampling distribution of the sample mean follows a normal distribution according to the Central Limit Theorem (for large n), and the Z-distribution provides accurate critical values for confidence intervals.

8. Confidence interval for the mean with unknown variance To calculate the confidence interval for the mean when the population variance is unknown, we use the t-distribution instead of the normal distribution. This accounts for the additional uncertainty in estimating the population variance from the sample.

9. Q. What is the 95% confidence interval for a sample mean of 50, with a sample standard deviation of 10 and a sample size of 25? Why do we use the t-distribution when σ is unknown? A: The t-distribution accounts for additional uncertainty due to estimating the population standard deviation from the sample. It is more spread out (has heavier tails) than the Z-distribution, especially for small sample sizes, and becomes closer to the normal distribution as the sample size increases.

10. Confidence interval for the variance To calculate the confidence interval for the variance of a population, you can use the chi-square distribution when the data is assumed to follow a normal distribution. The confidence interval is based on the sample variance and uses the chi-square critical values for the chosen confidence level.

11. Example: Let’s say we have a sample of size n=10 with a sample variance =25, and we want to compute the 95% confidence interval for the variance.