Step 1 :The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one. The z-score is the number of standard deviations a given data point is from the mean. Here, we are asked to find the probability that the z-score is greater than 1.77.
Step 2 :In a standard normal distribution, the total area under the curve is 1, representing the total probability. The area to the left of the z-score represents the cumulative probability up to that z-score. Therefore, to find the probability that z > 1.77, we need to subtract the cumulative probability up to z = 1.77 from 1.
Step 3 :We can use the cumulative distribution function (cdf) for the standard normal distribution. The cdf gives the probability that a random variable is less than or equal to a certain value. So, we can use it to find the cumulative probability up to z = 1.77, and then subtract that from 1 to get the probability that z > 1.77.
Step 4 :The cumulative probability up to z = 1.77 is 0.9616364296371288.
Step 5 :Subtracting this from 1 gives the probability that z > 1.77, which is 0.0384.
Step 6 :Final Answer: The probability that z > 1.77 in a standard normal distribution is \(\boxed{0.0384}\).