Chapter 7: The Normal Probability Distribution

7.1 Properties of the Normal Distributions

The normal probability distribution is a fundamental concept in statistics, describing how data values are distributed in many natural and social phenomena. This section introduces the properties of normal distributions, how to graph them, and how to interpret areas under the curve as probabilities.

Use the Uniform Probability Distribution

• Probability Density Function (pdf): A function that describes the likelihood of a random variable taking on a particular value. For a continuous random variable, the probability that it falls within a certain interval is the area under the pdf over that interval.

• Uniform Distribution: All outcomes in a given interval are equally likely. The pdf is constant over the interval.

Example: If a package is delivered between 10:00 am and 11:00 am, the probability it arrives between 10:15 and 10:30 is proportional to the length of that interval.

Formula for Uniform pdf:

for

Graph a Normal Curve

• Normal Distribution: A continuous probability distribution that is symmetric about the mean, with data near the mean more frequent in occurrence than data far from the mean.

• Histogram Approximation: As sample size increases, the histogram of sample means approaches the normal curve.

Example: The distribution of heights, test scores, and measurement errors often follow a normal distribution.

State the Properties of the Normal Curve

• The curve is symmetric about the mean .

• The mean, median, and mode are all equal and located at the center.

• The curve has inflection points at and .

• The area under the curve is 1 (total probability).

• As moves away from the mean, the curve approaches, but never touches, the horizontal axis.

• Empirical Rule (68-95-99.7 Rule):

  • About 68% of the area is within

  • About 95% of the area is within

  • About 99.7% of the area is within

Formula for Normal pdf:

Explain the Role of Area in the Normal Density Function

• The area under the normal curve between two values represents the probability that a randomly selected value falls within that interval.

• Probabilities are found by calculating the area under the curve using tables or technology.

Example: If the mean distance a baseball is hit is 400 ft with a standard deviation of 25 ft, the probability a ball travels more than 450 ft is the area under the curve to the right of 450.

7.2 Applications of the Normal Distributions

This section focuses on finding and interpreting areas under the normal curve, and using the normal model to solve real-world problems.

Find and Interpret the Area Under a Normal Curve

• To find probabilities, standardize the variable using the z-score:

• Use z-tables or technology to find the area (probability) corresponding to a z-score.

Example: If ball bearings have diameters normally distributed with mean 1.0 cm and standard deviation 0.02 cm, the probability a bearing has a diameter less than 0.98 cm is found by calculating the area to the left of .

Using the Normal Model to Find Percentiles

• The percentile rank of a value is the percentage of data values below it.

• To find the value corresponding to a given percentile, use the inverse normal function (z-tables or technology).

Example: Find the value of such that 90% of values are below it (the 90th percentile).

Find the Value of a Normal Random Variable

• Given a probability (area), find the corresponding value of using the z-score formula solved for $x$:

• This is useful for quality control and setting specification limits.

7.3 Assessing Normality

Assessing whether data are approximately normally distributed is important for many statistical methods. This section introduces normal probability plots as a graphical tool for this purpose.

Use Normal Probability Plots to Assess Normality

• A normal probability plot compares the ordered values of a dataset to the expected z-scores from a normal distribution.

• If the plot is approximately linear, the data are likely normally distributed.

• Steps to create a normal probability plot:

  1. Order the data from smallest to largest.

  2. Find the expected z-scores for each data point.

  3. Plot the data values against the expected z-scores.

Example: If the plot of sample means from a factory process is linear, the process is likely normal.

7.4 The Normal Approximation to the Binomial Probability Distribution

When certain conditions are met, the binomial distribution can be approximated by the normal distribution, simplifying probability calculations for large samples.

Approximate Binomial Probabilities Using the Normal Distribution

• Criteria: The approximation is appropriate if and , where is the number of trials and is the probability of success.

• Mean and Standard Deviation:

  • , where

• Continuity Correction: When approximating discrete binomial probabilities with the continuous normal distribution, add or subtract 0.5 to the value of as appropriate.

Example: If 8% of households buy pet gifts and 200 are sampled, the probability that more than 20 buy gifts can be approximated using the normal distribution.

Summary Table: Properties of the Normal Distribution

Additional info: The notes include examples using technology (StatCrunch), interpretation of normal probability plots, and practical applications such as quality control and percentiles. The continuity correction is essential when using the normal approximation for discrete distributions.