Exploring Standard Normal Distribution and Z-Scores

This video tutorial covers the standard normal distribution, explaining its bell curve shape, mean, and standard deviation. It introduces z-scores and the probability density function, emphasizing the empirical rule (68-95-99.7) for understanding data distribution. The tutorial includes example problems to illustrate these concepts, demonstrating how to calculate probabilities using z-scores and z-tables. It applies these principles to real-world scenarios, such as IQ scores and quality control in manufacturing, providing a comprehensive understanding of normal distribution and its applications.

What are the two important parameters of a normal distribution?

• Mean and standard deviation

• Mean and variance

• Median and mode

• Range and interquartile range

What is the z-score formula?

• Z = (X - mode) / variance

• Z = (X - mean) / variance

• Z = (X - median) / standard deviation

• Z = (X - mean) / standard deviation

What percentage of x values lie within one standard deviation of the mean according to the empirical rule?

• 68%

• 50%

• 95%

• 99.7%

What is the probability density function (PDF) used for in a normal distribution?

• To calculate the area under the curve

• To determine the probability of a specific value

• To find the standard deviation

• To calculate the mean

Using the empirical rule, what is the probability of finding an x value within three standard deviations of the mean?

• 50%

• 99.7%

• 95%

• 68%

What is the z-score for an x value of 80 if the mean is 100 and the standard deviation is 15?

• -1.5

• 1.33

• -1.33

• 1.5

How do you find the area under the curve for a z-score of 1.56 using the z-table?

• Multiply the z-score by the standard deviation

• Divide the z-score by the standard deviation

• Look up the value directly in the table

• Subtract the z-score from the mean

What is the probability of randomly selecting someone with an IQ score greater than 136 if the mean is 100 and the standard deviation is 15?

• 0.5

• 0.9918

• 0.25

• 0.0082

What IQ score corresponds to the 90th percentile if the mean is 100 and the standard deviation is 15?

• 115

• 120

• 119.2

• 125

What is the probability that the number of defective tires will be between 7 and 14 if the mean is 10 and the standard deviation is 3.13?

• 0.7312

• 0.2688

• 0.25

• 0.5