Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

Verified step by step guidance

1. Identify the given information: The graph shows a standard normal distribution with a shaded area of 0.5987 to the left of the unknown z-score. The mean of the standard normal distribution is 0, and the standard deviation is 1.
2. Understand the relationship between the area under the curve and the z-score: The area to the left of a z-score in the standard normal distribution corresponds to the cumulative probability for that z-score.
3. Use a z-table or statistical software: Look up the cumulative probability of 0.5987 in a z-table (or use statistical software) to find the z-score that corresponds to this cumulative area.
4. Interpret the z-table: Locate the value 0.5987 in the body of the z-table. The row and column headers corresponding to this value will give the z-score. If using software, input the cumulative probability to directly obtain the z-score.
5. Verify the result: Ensure that the z-score found matches the given area (0.5987) by checking the cumulative probability for the calculated z-score in the z-table or software.

Key Concepts

Z-Score

A z-score represents the number of standard deviations a data point is from the mean of a distribution. It is calculated by subtracting the mean from the data point and then dividing by the standard deviation. Z-scores are essential for understanding how a particular value compares to the overall distribution, especially in the context of normal distributions.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (average) and the standard deviation (spread). Many statistical methods assume normality, making it crucial for interpreting z-scores and areas under the curve.

Area Under the Curve

In the context of a normal distribution, the area under the curve represents the probability of a random variable falling within a certain range. The total area under the curve equals 1, and specific areas correspond to probabilities associated with z-scores. Understanding how to calculate and interpret these areas is vital for statistical analysis and hypothesis testing.