Q

QuestionStatistics

We need to find the value of the z-score for a 96% confidence level. In statistics, the z-score represents the number of standard deviations a given value is from the mean in a normal distribution. A 96% confidence level means that we're looking for the z-score that corresponds to the middle 96% of the data in a normal distribution.

The z-score can be calculated using the following formula: However, in this case, we're not looking for a specific z-score but rather the z-score that corresponds to the 96% confidence level.

The z-score and the confidence level are related through the properties of the standard normal distribution. In a standard normal distribution, the area under the curve represents the total probability, which is equal to 1. The area under the curve between the mean and a given z-score represents the probability of observing a value within that range. For a 96% confidence level, we want to find the z-score that corresponds to the middle 96% of the data. Since the total area under the curve is 1, we can calculate the desired z-score by subtracting the area that corresponds to the remaining 4% from 0.5 (which represents the middle 50% of the data).

To calculate the z-score, we first need to find the area that corresponds to the remaining 4%. Since the normal distribution is symmetric, the area to the left of -z is equal to the area to the right of z. Therefore, the area to the right of z is 2% (half of the remaining 4%). Using a standard normal distribution table or calculator, we can find that the area to the right of a z-score of 2.05 is approximately 0.0200 (2%). This means that the area to the left of - 2.05 is also approximately 0.0200. Now, we can calculate the z-score for the 96% confidence level by subtracting the area to the left of - 2.05 from 0.5: z = 0.5 - (- 2.05) = 2.55