for 95 out 0f 100 of the same size, a 95% confidence interval of a parameter will include the thrue value of that parameter
True
17. A normal distribution has a mean of 50 and a standard deviation of 4.
a. Compute the probability of a value between 44.0 and 55.0.
b. Compute the probability of a value greater than 55.0.
c. Compute the probability of a value between 52.0 and 55.0.
To understand why a 95% confidence interval will include the true value of a parameter in 95 out of 100 cases, we need to understand the concept of confidence intervals and hypothesis testing.
A confidence interval is an estimated range of values that is likely to contain the true value of a population parameter. In this case, the parameter could be a mean, proportion, or any other population parameter we are interested in estimating.
Typically, a confidence interval is constructed based on a sample from the population. The width of the interval is determined by the level of confidence chosen, which is commonly expressed as a percentage. In your case, a 95% confidence interval is used.
The interpretation of a 95% confidence interval is as follows - if we were to replicate the study 100 times and construct a confidence interval each time, approximately 95 of those intervals would include the true value of the parameter. The remaining 5 intervals may not cover the true value due to sampling variability.
The level of confidence you choose represents the probability or level of certainty that the interval contains the true value. In this case, 95% confidence level means that there is a 95% probability that the true value falls within the calculated interval.
It's important to note that a confidence interval does not tell us anything about a specific sample. It only provides information about the likelihood of intervals constructed using the same methodology and repeated sampling.
Therefore, based on statistical theory, we can say that for 95 out of 100 intervals constructed using the same methodology and a 95% confidence level, the true value of the parameter will be included in the interval.