Final answer:
To calculate probabilities and find z-scores for a standard normal random variable, one can use a z-table or calculator functions like invNorm. This applies to finding specific z-scores given a probability or calculating a z-score for a given value, like a test score in relation to a class mean and standard deviation.
Step-by-step explanation:
To compute probabilities involving a standard normal random variable z, we need to refer to the standard normal distribution table or use calculator functions. If we are given a probability and need to find the corresponding z-score, we can use the inverse normal function on a calculator or the z-table. For example, to find z0.01, which is the z-score with 0.01 area to the right, we use the invNorm function on a calculator or look up the value in a z-table that corresponds to an area of 0.99 to the left of the z-score.
Here is a specific answer to the provided question: z0.01 = 2.326, because the area under the normal density curve to the right of z0.01 is 0.01 and to the left is 0.99. If Susan scored 95 on her biology exam, and the class mean is 85 with a standard deviation of five, we would compute her z-score as follows: (95-85)/5 = 2. This means Susan scored two standard deviations above the mean.
The z-score helps us understand how many standard deviations an observation is from the mean. In the standard normal distribution, the mean is zero and the standard deviation is one. Therefore, a z-score directly indicates the number of standard deviations from the mean.