Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Verified step by step guidance

1. Step 1: Understand the problem. The graph represents a standard normal distribution with a mean of 0 and a standard deviation of 1. The shaded area to the left of the z-score corresponds to a cumulative probability of 0.3050.
2. Step 2: Recall the relationship between z-scores and cumulative probabilities. The z-score is the value on the horizontal axis of the standard normal distribution that corresponds to a given cumulative probability.
3. Step 3: Use a standard normal distribution table or a statistical software/tool to find the z-score that corresponds to a cumulative probability of 0.3050. This involves looking up the cumulative probability in the table and identifying the associated z-score.
4. Step 4: Note that the cumulative probability represents the area under the curve to the left of the z-score. Ensure that the table or tool you use is designed for the standard normal distribution.
5. Step 5: Verify your result by checking that the cumulative probability for the identified z-score matches 0.3050. This ensures accuracy in your calculation.

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the bell-shaped curve, which is symmetric about the mean. This distribution is crucial for statistical analysis as it allows for the calculation of probabilities and z-scores, which indicate how many standard deviations an element is from the mean.

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are essential for understanding how far away a particular score is from the average, allowing for comparisons across different datasets.

Area Under the Curve

In the context of the normal distribution, the area under the curve represents the probability of a random variable falling within a particular range. The total area under the curve equals 1, and specific areas correspond to probabilities associated with z-scores. For example, the shaded area in the provided graph indicates the probability of obtaining a z-score less than 0, which is 0.3050.