1. If we wish to have a 95% confidence interval, what would be the value of the confidence coefficient? & Consider a sample mean (xbar) of 46 calculated from a sample of size 51 drawn from a population of over 1000.
To determine the value of the confidence coefficient for a 95% confidence interval, we need to look up the corresponding value from a standard normal distribution table.
The confidence coefficient represents the level of confidence we have in our estimate, which is the probability that the true population parameter falls within the interval. In this case, we want a 95% confidence interval, which means we want to be 95% confident that the true population parameter falls within the interval.
Since we're dealing with a standard normal distribution, we can look up the value directly from the z-table. The critical value for a 95% confidence interval is typically 1.96. This means that approximately 95% of the values in a standard normal distribution fall within 1.96 standard deviations of the mean.
Now, let's calculate the confidence interval based on the given sample mean (x̄) and sample size (n) of 51:
First, we need to calculate the standard error of the mean (SE):
SE = σ / sqrt(n)
Since the population standard deviation (σ) is not given, we can estimate it using the sample standard deviation (s) and the formula sigma ≈ s (sample standard deviation is a good estimator of the population standard deviation when the sample size is reasonably large).
Assuming the sample standard deviation (s) is known, you can calculate the standardized test statistic (z-score) as follows:
z = (x̄ - μ) / (SE)
where μ is the population mean.
Finally, we can calculate the confidence interval using the formula:
CI = x̄ ± (z * SE)
I hope this helps! Let me know if you have any further questions.