Graphical Analysis In Exercises 11–16, determine whether the graph could represent a variable with a normal distribution. Explain your reasoning. If the graph appears to represent a normal distribution, estimate the mean and standard deviation.

Verified step by step guidance

1. Step 1: Observe the graph and determine its shape. A normal distribution is characterized by a symmetric bell-shaped curve, where the highest point (the peak) corresponds to the mean, and the curve tapers off equally on both sides.
2. Step 2: Check for symmetry. Analyze whether the graph is symmetric around a central value. In this case, the graph appears to be symmetric around x = 50, which suggests it could represent a normal distribution.
3. Step 3: Estimate the mean. The mean of a normal distribution is the value at the center of the graph, where the peak occurs. Based on the graph, the mean is approximately x = 50.
4. Step 4: Estimate the standard deviation. The standard deviation can be estimated by observing the spread of the graph. For a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Identify the points where the curve starts to taper off significantly, which appear to be around x = 48 and x = 52. The standard deviation is roughly half the distance between these points and the mean.
5. Step 5: Conclude whether the graph represents a normal distribution. Based on the symmetry and bell-shaped curve, the graph appears to represent a variable with a normal distribution. The mean is approximately 50, and the standard deviation can be estimated using the spread of the data.

Key Concepts

Normal Distribution

A 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). In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, and about 95% falls within two standard deviations.

Mean and Standard Deviation

The mean is the average value of a dataset, calculated by summing all values and dividing by the number of observations. The standard deviation measures the dispersion of data points around the mean, indicating how spread out the values are. In a normal distribution, the mean, median, and mode are all equal, and the standard deviation helps determine the width of the bell curve.

Graphical Analysis

Graphical analysis involves interpreting visual representations of data, such as histograms or density plots, to identify patterns, trends, and distributions. In the context of normal distribution, one looks for symmetry, a single peak (unimodal), and the characteristic bell shape. Analyzing the graph can help estimate the mean and standard deviation based on the shape and spread of the data.