6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:05 minutesProblem 5.2.19a
Textbook Question
Pregnancy Length Use the normal distribution in Exercise 15.
a. What percent of the new mothers had a pregnancy length of less than 290 days?
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Identify the key parameters of the normal distribution: the mean (μ) and the standard deviation (σ). These values should be provided in Exercise 15.a. If not explicitly given, refer to the problem context to find them.
Standardize the value of 290 days using the z-score formula: , where x = 290, μ is the mean, and σ is the standard deviation.
Once the z-score is calculated, use a standard normal distribution table (z-table) or statistical software to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents the proportion of the population with a pregnancy length of less than 290 days.
Interpret the cumulative probability as a percentage by multiplying it by 100. This percentage represents the proportion of new mothers with a pregnancy length of less than 290 days.
Verify the result by ensuring the z-score and cumulative probability calculations are consistent with the properties of the normal distribution (e.g., probabilities should be between 0 and 1).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is widely used in statistics to represent real-valued random variables whose distributions are not known. In the context of pregnancy length, it helps in understanding how pregnancy durations are distributed around the average.
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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. In this question, calculating the Z-score for a pregnancy length of 290 days will help determine how many standard deviations this length is from the average, which is essential for finding the corresponding percentile.
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Percentile
A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, if a pregnancy length is at the 30th percentile, it means that 30% of pregnancies are shorter than this length. In this question, determining the percentile for a pregnancy length of less than 290 days will provide insight into how common or rare this length is among new mothers.
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