Determine the level of the confidence given the confidence coefficient z(α/2) for the following situation:
z(α/2) = 1.96
The level of confidence can be determined by using the confidence coefficient z(α/2).
In this case, z(α/2) is given as 1.96.
The confidence level is related to z(α/2) through the standard normal distribution, which specifies the area under the curve to the left of z(α/2).
Since z(α/2) = 1.96 corresponds to an area of 0.975 (from a z-table or calculator), we can determine the level of confidence by subtracting this value from 1 and multiplying by 100:
Confidence level = (1 - 0.975) * 100 = 2.5%
Therefore, the level of confidence associated with z(α/2) = 1.96 is 97.5%.
To determine the level of confidence given the confidence coefficient z(α/2) = 1.96, we need to refer to the standard normal distribution table.
The confidence coefficient, z(α/2), represents the number of standard deviations from the mean to the area under the normal distribution curve that corresponds to a specific confidence level. In the case of a two-tailed test, α/2 represents the significance level divided by 2.
1. Start by locating the z-score of 1.96 in the standard normal distribution table. The standard normal distribution table provides the area under the curve to the left of each z-score.
2. Once you locate 1.96 in the table or calculate it using statistical software, you will find that the area to the left of 1.96 is 0.975 (or 97.5% if expressed as a percentage).
3. Since we are considering a two-tailed test, we need to find the area in both tails of the distribution. Therefore, we subtract 0.975 from 1 to get 0.025 (or 2.5% if expressed as a percentage). This 0.025 represents the combined area in both tails of the distribution.
4. To determine the level of confidence, we need to consider the area in one tail only. Thus, we divide the combined area (0.025) by 2. This gives us 0.0125 (or 1.25% if expressed as a percentage).
5. Finally, we subtract this value from 0.5 (50% or the area under the curve corresponding to the mean) to get the level of confidence. Therefore, the level of confidence is 0.5 + 0.0125 = 0.5125 (or 51.25% if expressed as a percentage).
So, for a confidence coefficient of z(α/2) = 1.96, the level of confidence is approximately 51.25%.