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Assume that the lengths of pregnancy for humans is approximately Normally distributed, with a mean of 267 days and a standard deviation of 10 days. Use the Empirical Rule to answer the following questions. Do not use the technology or the Normal table. Begin by labeling the horizontal axis of the graph with lengths, using the given mean and standard deviation. Three of the entries are done for you. a. Roughly what percentage of pregnancies last more than 267 days? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) b. Roughly what percentage of pregnancies last between 267 and 277 days? i. \(34 \%\) iii. \(2.5 \%\) ii. \(17 \%\) iv. \(50 \%\) c. Roughly what percentage of pregnancies last less than 237 days? i. almost all iii. \(34 \%\) ii. \(50 \%\) iv. about \(0 \%\) d. Roughly what percentage of pregnancies last between 247 and 287 days? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) e. Roughly what percentage of pregnancies last longer than 287 days? i. \(34 \%\) iii. \(2.5 \%\) ii. \(17 \%\) iv. \(50 \%\) f. Roughly what percentage of pregnancies last longer than 297 days? i. almost all iii. \(34 \%\) ii. \(50 \%\) iv. about \(0 \%\)

Step by step solution

01

Percent of pregnancies lasting more than 267 days

Pregnancies lasting more than 267 days are those exceeding the mean. According to the the Empirical Rule, half of the data lies above the mean and half lies below. Therefore, roughly 50% of pregnancies last more than 267 days. So, the answer is iv. \(50 \% \)

02

Percent of pregnancies lasting between 267 and 277 days

Pregnancies lasting between 267 and 277 days fall within one standard deviation above the mean. In a normal distribution, approximately 34% of the data lies between the mean and one standard deviation above the mean. So, the answer is i. \(34 \% \)

03

Percent of pregnancies lasting less than 237 days

Pregnancies lasting less than 237 days fall within three standard deviations below the mean. In a normal distribution, approximately 0.15% of the data lies three standard deviations below the mean. So, the answer is iv. about \(0 \% \)

04

Percent of pregnancies lasting between 247 and 287 days

Pregnancies lasting between 247 and 287 days fall within two standard deviations of the mean. In a normal distribution, approximately 95% of the data lies within two standard deviations of the mean. So, the answer is ii. \(95 \% \)

05

Percent of pregnancies lasting longer than 287 days

Pregnancies lasting longer than 287 days fall within two standard deviations above the mean. In a normal distribution, approximately 2.5% of the data lies two standard deviations above the mean. So, the answer is iii. \(2.5 \% \)

06

Percent of pregnancies lasting longer than 297 days

Pregnancies lasting longer than 297 days fall within three standard deviations above the mean. In a normal distribution, approximately 0.15% of the data lies three standard deviations above the mean. So, the answer is iv. about \(0 \% \)

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Key Concepts

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

Normal Distribution

The normal distribution, often referred to as the bell curve, is a fundamental concept in statistics that describes how values of a random variable spread out. The majority of values cluster around a central point—the mean—with fewer and fewer values as we move further away. This creates a symmetric, bell-shaped curve when plotted graphically. Values closer to the mean are more frequent, and those far from the mean are less so.

In the context of pregnancy lengths, we can imagine plotting the number of pregnancies against days to see their distribution. Most would center around the mean with diminishing frequencies as we move toward shorter or longer durations. The Empirical Rule is a handy guideline for normal distributions, indicating that approximately 68% of the data falls within one standard deviation, 95% within two, and almost all (99.7%) within three standard deviations from the mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that most data points are close to the mean, while a high standard deviation shows more spread out data. In the context of pregnancy lengths, a standard deviation of 10 days means that many pregnancies last about 10 days less or more than the average duration of 267 days.

Understanding standard deviation is crucial when applying the Empirical Rule. If we know the standard deviation is 10 days, we can expect that a significant number of pregnancies will fall within the range of 257 to 277 days (which is the mean minus one standard deviation to the mean plus one standard deviation).

Statistics in Pregnancy

In the realm of obstetrics, statistics play a vital role in understanding what to expect during the course of a pregnancy. For healthcare providers and expecting parents, knowing the typical distribution of pregnancy lengths helps in planning and management. Being aware that a certain percentage of pregnancies fall within certain duration windows can alleviate concerns or, conversely, signal when there may be a need for medical intervention.

For instance, if we know that roughly 95% of pregnancies last between 247 and 287 days, a pregnancy reaching beyond this may be flagged for closer observation.

Percentiles in Normal Distribution

Percentiles are useful for understanding where a particular value falls within a distribution. For example, if a baby is born at a length that is at the 90th percentile, this means that they are longer than 90% of babies in the reference set.

Using the Empirical Rule, we can easily determine percentiles in a normal distribution without complex calculations. This is especially helpful in settings like tracking pregnancy lengths where precise data may not be available. Knowing the range of percentiles gives us a quick way to identify the stages of pregnancy and address any concerns at specific points during its progression.