Suppose that, based on a simple random sample of n=50 English Premier League soccer matches, we wish to calculate a 94% confidence interval for the population mean number of offside violations per game. For this confidence interval, the critical value +/- z* could be calculated from:
a. the 2.5th percentile of a N(0,1) distribution
b. the 3rd percentile of a N(0,1) distribution
c. the 6th percentile of a N(0,1) distribution
d. the 94th percentile of a N(0,1) distribution
The correct answer is a. the 2.5th percentile of a N(0,1) distribution.
To calculate a confidence interval, we need to find the critical value, also known as the Z-score, which corresponds to the desired level of confidence. In this case, the confidence level is 94%, which means there is a 94% probability that the true population mean falls within the calculated interval.
To find the Z-score for a 94% confidence interval, we subtract the confidence level from 100% to get the percentage in the tails of the distribution. Since the distribution is symmetric, we divide this tail percentage in half, resulting in 3% in each tail.
The critical value is then the Z-score that corresponds to a cumulative probability of 3% or a percentile of 2.5%. Therefore, we need to calculate the 2.5th percentile of a standard normal distribution (N(0,1)) to find the critical value for our confidence interval.
Why is it not the 3rd percentile?
I apologize for the confusion in my previous response. You are correct; the correct answer is b. the 3rd percentile of a N(0,1) distribution.
When calculating a confidence interval, we need to determine the critical value, which corresponds to the desired level of confidence. In this case, a 94% confidence interval means that 94% of the data falls within the interval, leaving 6% in the tails of the distribution.
Since the distribution is symmetric, we divide this tail percentage in half, resulting in 3% in each tail. To find the Z-score that corresponds to a cumulative probability of 3% or a percentile of 3rd, we would need to calculate the 3rd percentile of a standard normal distribution (N(0,1)) to determine the critical value for our confidence interval. Once again, I apologize for the incorrect information in my previous response.
To calculate a confidence interval, we need to determine the critical value, denoted as z* in this case. The critical value is the number of standard deviations away from the mean that corresponds to the desired confidence level.
In this scenario, we want to calculate a 94% confidence interval. The remaining percentage, 100% - 94% = 6%, will be divided equally between the two tails of the distribution. To find the critical value, we need to find the value on a standard normal distribution that corresponds to a cumulative probability of 0.5 + (6%/2) = 0.53.
To find this value, we can use a Z-table or normal distribution calculator. The value corresponding to a cumulative probability of 0.53 is approximately 0.06.
Since the standard normal distribution has a mean of 0 and a standard deviation of 1, the critical value can be calculated as z* = 0.06.
To summarize, the correct option for calculating the critical value is c. the 6th percentile of a N(0,1) distribution.