Final answer:
To find the area under the standard normal curve, calculate the z-score and use a z-table or statistical software. The standard normal distribution has a mean of 0 and a standard deviation of 1, and is used to find probabilities for normally distributed variables.
Step-by-step explanation:
To find an indicated area under the standard normal curve, we first need to calculate the z-score for the given probability or percentile. Once we have the z-score, we use a z-table, calculator, or software to find the area under the curve to the left of the z-score. For instance, to find the 90th percentile, we look for a z-score where the area to the left under the curve is 0.9. According to most z-tables, the z-score corresponding to an area of 0.9 is approximately 1.28. This z-score represents the point on the curve where 90% of the values lie to the left and 10% to the right.
The standard normal distribution is symmetric about the mean, and has a mean (μ) of 0 and standard deviation (σ) of 1. The total area under the curve equals to one, representing all possible outcomes. This distribution is used as a reference to find probabilities and percentile ranks for normally distributed variables.
In practical applications like testing a sample mean without a known population standard deviation, we would use a Student's t-distribution. However, for a normal distribution, once the z-score is identified, it is straightforward to find the required areas under the curve.