Determine the level of the confidence given the confidence coefficient z(α/2) for the following situation: z(α/2) = 1.96
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for Z = ±1.96. Hint: It is the area between Z and the mean.
To determine the level of confidence, we need to understand what the confidence coefficient represents. In statistics, the confidence coefficient, denoted as z(α/2), is associated with a particular level of confidence in a confidence interval.
In this case, z(α/2) is given as 1.96. To determine the level of confidence, we need to find the corresponding level of confidence associated with this value.
The z(α/2) value represents the critical value on a standard normal distribution for a two-tailed test. It is essentially the number of standard deviations away from the mean that corresponds to the desired level of confidence.
For a two-tailed test, where we want to capture the area in both tails of the distribution, we divide the desired level of confidence (α) by 2. In this case, α/2 represents the significance level divided by 2.
To find the level of confidence, we need to use a standard normal distribution table (also known as a Z-table) or a calculator that can provide the corresponding level of confidence for a given z-score.
Looking up the value of 1.96 in a Z-table or using a calculator, we find that it corresponds to a level of confidence of approximately 0.9750 or 97.50%.
Therefore, the level of confidence associated with z(α/2) = 1.96 is approximately 97.50%.
The confidence level can be determined by using the confidence coefficient formula:
Confidence level = 1 - α
Where α is the significance level.
In this case, the confidence coefficient z(α/2) is given as 1.96.
Since α/2 corresponds to the tail probability, we need to find the area under the standard normal curve to the right of -1.96.
Using a standard normal distribution table, we find that the area to the right of -1.96 is 0.025.
Since the confidence level is equal to 1 - α, we subtract the tail probability from 1 to determine the confidence level:
Confidence level = 1 - 0.025 = 0.975
Therefore, the level of confidence is 97.5%, since the corresponding confidence level is 0.975 or 97.5%.