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Chapter 6: Problem 14

Length of Pregnancy 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 \(0 \%\)

Short Answer

Expert verified
a) iv. 50% b) i. 34% c) iv. about 0% d) ii. 95% e) iii. 2.5% f) iv. about 0%

Step by step solution

01

Understand the Normal Distribution

A normal distribution is shaped like a bell curve. The peak represents the mean value. In this case, 267 days. The standard deviation is the measure of dispersion and it tells how spread out the distribution is. Here, it is 10 days. So, 68% of the values lie within one standard deviation, 267-10=257 to 267+10=277 days.
02

Apply the Empirical Rule

a) The percentage of pregnancies lasting more than the mean (267 days) would be around 50%. Because on a normal distribution, 50% of the values lie above the mean. \n b) The percentage of pregnancies lasting between 267 and 277 days will be approximately 34%. This is because, according to the Empirical Rule, 68% of data falls within one standard deviation from the mean, meaning 34% falls on either side, so between the mean and mean + one standard deviation would cover roughly 34%. \n c) The percentage of pregnancies lasting less than 237 days will be about 0%. This is 3 standard deviations less than the mean, which covers 99.7% of data according to the empirical rule, hence almost none last less than this. \n d) The percentage of pregnancies lasting between 247 and 287 days will be roughly 95% - it is within 2 standard deviations. \n e) The percentage of pregnancies lasting longer than 287 days will be approximately 2.5%. It is within 2 standard deviations which is covering 95% of pregnancies and hence 5% are either below 257 days or above 287 days. Thus, approximately 2.5% will be above 287 days. \n f) The percentage of pregnancies lasting longer than 297 days will be about 0%. As per empirical rule, 99.7% of pregnancies last between 237 days and 297 days, thus almost no pregnancies last longer than 297 days.
03

Review and Conclusion

Review the percentages calculated for each of the given situations. These percentages are rough estimates based on the empirical rule and the given mean and standard deviation. The understanding of normal distribution and empirical rule is critical here, as this rule can help in answering many questions related to distributions in various real-life scenarios.

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Key Concepts

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

Normal Distribution
Imagine a bell curve 鈥� this is what statisticians call a 'normal distribution.' It's a graph that beautifully represents data which is clustered around a central value with no bias towards left or right. In our example, the length of human pregnancies is clustered around 267 days, which is the mean, or the high point of our bell. This distribution is symmetric, meaning that it mirrors itself on either side of the mean.

Why does this matter? Because lots of real-world data sets, including test scores or heights, tend to follow this pattern. Knowing this lets us predict and understand patterns in the world around us. For instance, with our bell curve peaking at 267 days, we can anticipate that most births will occur near this point and progressively fewer earlier or later than this.
Standard Deviation
In the symphony of statistics, the standard deviation is the measurement of the diversity in the tune. It tells us how spread out our data is on the normal distribution curve. Think of it as an indication of variability around the mean. For the length of pregnancies, a standard deviation is 10 days. This means that if we move 10 days away from our mean of 267 days in either direction, we cover a lot of ground 鈥� specifically, 68% of our data.

Why is standard deviation such a big deal? It gives us a numerical way to express uncertainty. For example, saying that a typical pregnancy lasts 267 plus or minus 10 days provides much more information than just the average. It can be applied in various fields like finance to gauge market volatility, in quality control to measure process variations, and in the health sector to assess variability in health-related outcomes.
Statistics in Real-Life Scenarios
Statistics are your hidden superpower when it comes to making sense of the world. They turn vague notions into actionable insights. Case in point: our pregnancy durations. By understanding the normal distribution and standard deviations, healthcare professionals can estimate time frames for most pregnancies and plan accordingly.

In business, this can mean forecasting sales and inventory requirements. In meteorology, predicting the likelihood of weather events. Statistics are crucial in decision-making and in predicting outcomes in nearly every field you can imagine. It鈥檚 like having a crystal ball, but instead of magic, we use data and rules like the empirical rule to see into the future 鈥� or at least make educated guesses. Knowing how to use these tools lets us navigate the uncertainties of life with more confidence.

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Most popular questions from this chapter

According to the American Veterinary Medical Association, \(36 \%\) of Americans own a dog. a. Find the probability that exactly 4 out of 10 randomly selected Americans own a dog. b. In a random sample of 10 Americans, find the probability that 4 or fewer own a dog.

Medical school graduates who want to become doctors must pass the U.S. Medical Licensing Exam (USMLE). Scores on this exam are approximately Normal with a mean of 225 and a standard deviation of \(15 .\) Use the Empirical Rule to answer these questions. a. Roughly what percentage of USMLE scores will be between 210 and 240 ? b. Roughly what percentage of USMLE scores will be below 210 ? c. Roughly what percentage of USMLE scores will be above 255 ?

Determine whether each of the following variables would best be modeled as continuous or discrete: a. Number of girls in a family b. Height of a tree c. Commute time d. Concert attendance

Toronto drivers have been going to small towns in Ontario (Canada) to take the drivers' road test, rather than taking the test in Toronto, because the pass rate in the small towns is \(90 \%\), which is much higher than the pass rate in Toronto. Suppose that every day, 100 people independently take the test in one of these small towns. a. What is the number of people who are expected to pass? b. What is the standard deviation for the number expected to pass? c. After a great many days, according to the Empirical Rule, on about \(95 \%\) of these days the number of people passing the test will be as low as and as high as d. If you found that on one day, 89 out of 100 passed the test, would you consider this to be a very high number?

New York City's mean minimum daily temperature in February is \(27^{\circ} \mathrm{F}\) (http://www.ny.com). Suppose the standard deviation of the minimum temperature is \(6^{\circ} \mathrm{F}\) and the distribution of minimum temperatures in February is approximately Normal. What percentage of days in February has minimum temperatures below freezing \(\left(32^{\circ} \mathrm{F}\right) ?\)
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