The heights of 10 -year-old males are normally distributed with mean μ=55.9 inches and σ=5.7 inc

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The heights of 10 -year-old males are normally distributed...

The heights of 10 -year-old males are normally distributed with mean $\mu=55.9$ inches and $\sigma=5.7$ inches. (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of $10-$ year-old males who are less than 46.5 inches tall. (c) Suppose the area under the normal curve to the left of $X=46.5$ is $0.0496 .$ Provide two interpretations of this result.

The heights of 10 -year-old males are normally distributed with mean $\mu=55.9$ inches and $\sigma=5.7$ inches.
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of $10-$ year-old males who are less than 46.5 inches tall.
(c) Suppose the area under the normal curve to the left of $X=46.5$ is $0.0496 .$ Provide two interpretations of this result.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean, which determines the location of the center of the curve, and the standard deviation, which determines the spread of the distribution. This concept is central to many areas in statistics and probability because many real-world phenomena tend to follow this pattern under certain conditions.

Mean

The mean of a distribution is the expected value or average of the data. In the context of a normal distribution, the mean is the central point around which the data are symmetrically distributed. It provides a measure of central tendency, and its value determines the horizontal position of the peak of the normal curve.

Standard Deviation

The standard deviation is a measure of the dispersion or spread of the data around the mean. In a normal distribution, a smaller standard deviation indicates that the data points are closely clustered around the mean, while a larger standard deviation implies they are more spread out. It is a crucial parameter that affects the shape and width of the normal curve.

Graphical Representation of the Normal Curve

Drawing the normal curve involves sketching the bell-shaped curve along a horizontal axis, labeling the mean at the center, and indicating one or more standard deviations from the mean. This graphical representation helps in visualizing the probability distribution of the data and understanding how probability is assigned to different ranges of values.

Probability and Area Under the Curve

The area under the normal curve represents the probability of the variable falling within a certain range. For example, the area to the left of a specific value on the horizontal axis represents the probability that a randomly selected observation will be less than that value. Because the total area under the curve is equal to 1, these areas can be directly interpreted as probabilities or percentages of the population.

Interpretation of Probabilistic Results

Interpreting specific areas under the normal curve, such as an area of 0.0496 to the left of a certain value, involves recognizing that this value represents the probability (or percentage) of observations falling below that threshold. This interpretation can be presented from multiple perspectives, such as the chance that a randomly chosen individual has a value below the given threshold, or that approximately 4.96% of the population falls below that value.

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