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Chapter 3: Problem 36

Distributions of gestation periods (lengths of pregnancy) for humans are roughly bell-shaped. The mean gestation period for humans is 272 days, and the standard deviation is 9 days for women who go into spontaneous labor. Which is more unusual, a baby being born 9 days early or a baby being born 9 days late? Explain.

Short Answer

Expert verified
Neither a baby being born 9 days early nor a baby being born 9 days late is more unusual. They both fall within one standard deviation from the mean in the distribution, making them both equally 'usual'.

Step by step solution

01

Calculate Early and Late Gestation Periods

First, calculate the gestation periods for both scenarios. The early gestation period is found by subtracting 9 days from the mean gestation period. Similarly, the late gestation period is calculated by adding 9 days to the mean. This gives us \(272 - 9 = 263\) days for the early period and \(272 + 9 = 281\) days for the late period.
02

Compare with Mean and Standard Deviation

Next, compare these calculated gestation periods with the mean and standard deviation. Both 263 and 281 fall within one standard deviation (9 days) of the mean (272 days). This means both are considered 'usual' in a bell-shaped distribution.
03

Draw Conclusion

Since both calculated gestation periods fall within one standard deviation from the mean, neither is considered more unusual than the other. In a bell-shaped distribution, it is perceived as equally 'usual' for a baby to be born 9 days early as it is for a baby to be born 9 days late.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Bell-Shaped Distributions
In statistics, a bell-shaped distribution is often referred to as a normal distribution. This is important because many natural phenomena, like human gestation periods, follow this pattern. The curve is symmetrical, meaning it looks like a bell. Most of the data points lie close to the mean, or average value, with probabilities tapering off equally on either side.

This shape helps us understand how occurrences are spread out around a central value. In our case, it means that most babies are born close to the average gestation period. This predictability makes it easier to determine if a gestation period is usual or unusual.
Mean - The Average Gestation Period
The mean is the mathematical average of a set of numbers. In the context of gestation periods, the mean is 272 days. The mean is crucial because it represents the typical length of a pregnancy for women who experience spontaneous labor.

Calculating the mean involves adding up all gestation period data and dividing by the number of observations. This central point helps in understanding the "usual" time frame for human pregnancies and forms the basis for comparing any deviations.
Standard Deviation - Measuring the Spread
Standard deviation is a measure that describes how spread out the numbers in a data set are around the mean. For gestation periods, the standard deviation is 9 days. This tells us that most pregnancies will vary by about 9 days either side of the mean 272 days.

A smaller standard deviation means that the values are closer to the mean, indicating a more tightly clustered set of data points. Conversely, a larger standard deviation suggests more variability. In our context, pregnancies falling within one standard deviation (between 263 and 281 days) are considered usual, falling within the norms of the bell-shaped distribution.
Spontaneous Labor and its Predictability
Spontaneous labor refers to labor that begins on its own without medical intervention. Understanding the statistical patterns related to it can be very insightful. Since gestation periods for spontaneous labor follow a normal distribution, we can predict their timing within certain limits.

This predictability allows us to determine what is usual or unusual. For example, given the normal distribution, both a baby born 9 days early or 9 days late are considered usual, as these timings fall within one standard deviation of the mean. This shows how the concept of spontaneous labor interacts with statistical measures to help manage expectations around birth timing.

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Most popular questions from this chapter

The mean weight gain for women during a full-term pregnancy is \(30.2\) pounds. The standard deviation of weight gain for this group is \(9.9\) pounds, and the shape of the distribution of weight gains is symmetric and unimodal. a. State the weight gain for women one standard deviation below the mean and for one standard deviation above the mean. b. Is a weight gain of 35 pounds more or less than one standard deviation from the mean?

In 2017 a pollution index was calculated for a sample of cities in the western states using data on air and water pollution. Assume the distribution of pollution indices is unimodal and symmetric. The mean of the distribution was \(43.0\) points with a standard deviation of \(11.3\) points. a. What percentage of western cities would you expect to have a pollution index between \(31.7\) and \(54.3\) points? b. What percentage of western cities would you expect to have a pollution index between \(20.4\) and \(65.6\) ? c. The pollution index for San Jose in 2017 was \(51.9\) points. Based on this distribution, was this unusually high? Explain.

Name two measures of the variation of a distribution, and state the conditions under which each measure is preferred for measuring the variability of a single data set.

Quantitative SAT scores have a mean of 500 and a standard deviation of 100 , while ACT scores have a mean of 21 and a standard deviation of \(5 .\) Assuming both types of scores have distributions that are unimodal and symmetric, which is more unusual: a quantitative SAT score of 750 or an ACT score of 28 ? Show your work.

The mean birth length for U.S. children born at full term (after 40 weeks) is \(52.2\) centimeters (about \(20.6\) inches). Suppose the standard deviation is \(2.5\) centimeters and the distributions are unimodal and symmetric. a. What is the range of birth lengths (in centimeters) of U.S.-bom children from one standard deviation below the mean to one standard deviation above the mean? b. Is a birth length of 54 centimeters more than one standard deviation above the mean?
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