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Babies born after 40 weeks gestation have a mean length of \(52.2\) centimeters (about \(20.6\) inches). Babies born one month early have a mean length of \(47.4 \mathrm{~cm}\). Assume both standard deviations are \(2.5 \mathrm{~cm}\) and the distributions are unimodal and symmetric. (Source: www.babycenter.com) a. Find the standardized score (z-score), relative to all U.S. births, for a baby with a birth length of \(45 \mathrm{~cm}\). b. Find the standardized score of a birth length of \(45 \mathrm{~cm}\) for babies born one month early, using \(47.4\) as the mean. c. For which group is a birth length of \(45 \mathrm{~cm}\) more common? Explain what that means.

Short Answer

Expert verified
The z-score for a baby with a length of 45cm born after 40 weeks is \(-2.88\). The z-score for a baby with a length of 45cm born one month early is \(-0.96\). Given the results, a length of 45cm is more common among babies born one month early, because this z-score is smaller in absolute value which means it is closer to the group's mean length.

Step by step solution

01

Calculate the z-score for babies born after 40 weeks

The formula for calculating the z-score is as following: \(z = \frac{{X - \mu}}{{\sigma}}\), where X is the score, \(\mu\) is the mean and \(\sigma\) is the standard deviation. Substitute \(X = 45cm\), \(\mu = 52.2cm\) and \(\sigma = 2.5cm\) into the formula. This gives: \(z = \frac{{45 - 52.2}}{{2.5}}\)
02

Calculate the z-score for babies born one month early

Applying the formula again, the score X is 45cm, but this time the mean \(\mu\) is 47.4cm (since this is for babies born one month early) and the standard deviation \(\sigma\) is still 2.5cm. So, \(z = \frac{{45 - 47.4}}{{2.5}}\) will give the z-score for a baby that is 45cm long, born one month early.
03

Compare the z-scores

When the calculation is finished, compare the z-scores obtained from step 1 and step 2. The group with the smaller absolute value of z-score will be the one for which a length of 45cm is more common. The meaning of the smaller z-score can be concluded as: a length of 45cm is closer to the mean length for this group of babies, compared to the other group.

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Key Concepts

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

Birth Length Statistics
In birth statistics, one interesting measure is the mean birth length. For example, babies born after 40 weeks of gestation have an average length of 52.2 cm. This statistic provides a benchmark to understand what the typical baby looks like at full term.
For babies born one month early, the average length is shorter at 47.4 cm. Comparing these two averages helps to highlight how gestational age influences birth size.
Average lengths like these are calculated from large samples of births to represent a population norm, offering valuable insights for healthcare and parental expectations.
  • Full-term babies generally measure around 52.2 cm on average.
  • Babies born a month early measure approximately 47.4 cm on average.
Understanding these norms is crucial for interpreting individual birth lengths and can indicate whether further investigation is needed for unusually short or long measurements.
Standard Deviation
Standard deviation is a key concept in birth length statistics that tells us how spread out the lengths are in a set of data. Here, both groups of babies (those born after 40 weeks and those born a month early) have a standard deviation of 2.5 cm.
This measure shows that most full-term babies' lengths fall within 2.5 cm of the mean of 52.2 cm. Similarly, most premature babies' lengths will be within 2.5 cm of the 47.4 cm mean.
  • Indicates how much the birth lengths vary from the average.
  • A standard deviation of 2.5 cm means that the lengths are fairly consistent.
In everyday terms, a standard deviation gives us a way to see how typical any one birth length might be compared to the average. A small standard deviation, like 2.5 cm, means there's less variety in the birth lengths, suggesting a more uniform distribution of sizes.
Unimodal Distribution
A unimodal distribution in the context of birth length statistics signifies that there is one peak in the data, indicating a single most common length. Both groups of babies have unimodal distributions, suggesting that there's a predominant birth length within each group.
This shape is often symmetric as well, meaning the lengths are equally spread on either side of the peak. In practical terms, this means most babies' lengths are close to the average, with fewer lengths at the extremes.
  • Reflects a single central tendency - most babies are around the average length.
  • Symmetric distribution shows equality in spread above and below the mean.
This pattern helps healthcare professionals understand and predict the likely range of baby lengths, providing a useful framework for identifying outliers who might need additional care or monitoring.

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