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Chapter 3: Problem 5

Babies born after a gestation period of 32-35 weeks have a mean weight of 2600 grams and a standard deviation of 660 grams. Babies born after a gestation period of 40 weeks have a mean weight of 3500 grams and a standard deviation of 470 grams. Suppose a 34 -week gestation period baby weighs 2400 grams and a 40 -week gestation period baby weighs 3300 grams. What is the \(z\) -score for the 34 -week gestation period baby? What is the \(z\) -score for the 40 -week gestation period baby? Which baby weighs less relative to the gestation period?

Short Answer

Expert verified
The Z-score for the 34-week baby is -0.303, and for the 40-week baby, it is -0.426. The 40-week baby weighs less relative to the gestation period.

Step by step solution

01

Understand the Z-score Formula

The Z-score formula helps to determine how many standard deviations an element is from the mean. The formula for the Z-score is: \[ z = \frac{X - \mu}{\sigma} \] where: - \(X\) is the value of the element - \(\mu\) is the mean - \(\sigma\) is the standard deviation.
02

Identify the Values for the 34-week Gestation Period Baby

For the 34-week gestation period baby: - Mean (\(\mu\)) = 2600 grams - Standard Deviation (\(\sigma\)) = 660 grams - Baby's Weight (\(X\)) = 2400 grams
03

Calculate the Z-score for the 34-week Gestation Period Baby

Using the provided formula for the Z-score, substitute the values: \[ z = \frac{2400 - 2600}{660} \] Calculate: \[ z = \frac{-200}{660} \approx -0.303 \]
04

Identify the Values for the 40-week Gestation Period Baby

For the 40-week gestation period baby: - Mean (\(\mu\)) = 3500 grams - Standard Deviation (\(\sigma\)) = 470 grams - Baby's Weight (\(X\)) = 3300 grams
05

Calculate the Z-score for the 40-week Gestation Period Baby

Using the same Z-score formula, substitute the values: \[ z = \frac{3300 - 3500}{470} \] Calculate: \[ z = \frac{-200}{470} \approx -0.426 \]
06

Compare the Z-scores

The Z-score for the 34-week gestation period baby is approximately -0.303, and the Z-score for the 40-week gestation period baby is approximately -0.426. Since -0.426 is further from zero than -0.303, the 40-week gestation period baby weighs less relative to its gestation period compared to the 34-week gestation period baby.

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Key Concepts

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

standard deviation
Standard deviation is a measure of how spread out the values in a data set are. It shows the average distance of each data point from the mean. In the context of our exercise, the standard deviation helps us understand the variability in baby weights for different gestation periods.
For babies born after 32-35 weeks, the standard deviation of weight is 660 grams, meaning most baby weights will fall within 660 grams of the mean weight of 2600 grams. For babies born after 40 weeks, the standard deviation is smaller, at 470 grams, indicating that these baby weights vary less around the mean of 3500 grams.
mean
The mean, also known as the average, is the sum of all values divided by the number of values. It represents the central value of a dataset. To better understand it, let's underscore the mean weights in the exercise:
- The mean weight for babies born after a 32-35 week gestation period is 2600 grams.
- The mean weight for babies born after a 40-week gestation period is 3500 grams.
These means give us a central value around which the weights of babies are distributed. By comparing a baby's weight to these means, we can understand how much that weight deviates from the typical weight for their gestation period.
gestation period
The gestation period is the length of time a baby develops in the womb before being born. This time can significantly impact a baby's weight.
In our exercise:
- A baby born after 32-35 weeks (prematurely) has a different expected average weight and variability (mean and standard deviation) compared to a baby born at full term (40 weeks).
Understanding the gestation period is key to contextualizing the mean and standard deviation of baby weights, and how a baby's weight compares to others born at a similar stage.
normal distribution
A normal distribution is a probability distribution that is symmetric around the mean. This shape of the distribution implies that most data falls close to the mean, with fewer values appearing as you move away.
In the context of our exercise:
- The weights of babies born after both 32-35 weeks and 40 weeks are assumed to follow a normal distribution.
This assumption lets us use the Z-score to find how far a particular baby's weight deviates from the mean in terms of standard deviations. The normal distribution is bell-shaped, indicating that most babies' weights are clustered around the mean, with few babies weighing much more or much less than the average.

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Most popular questions from this chapter

Compute the sample standard deviation of the following test scores: 78,78,78,78 . What can be said about a data set in which all the values are identical?

The following data represent exam scores in a statistics class taught using traditional lecture and a class taught using a "flipped" classroom. The "flipped" classroom is one where the content is delivered via video and watched at home, while class time is used for homework and activities. $$ \begin{array}{llllllll} \hline \text { Traditional } & 70.8 & 69.1 & 79.4 & 67.6 & 85.3 & 78.2 & 56.2 \\\ & 81.3 & 80.9 & 71.5 & 63.7 & 69.8 & 59.8 & \\ \hline \text { Fipped } & 76.4 & 71.6 & 63.4 & 72.4 & 77.9 & 91.8 & 78.9 \\ & 76.8 & 82.1 & 70.2 & 91.5 & 77.8 & 76.5 & \end{array} $$ (a) Which course has more dispersion in exam scores using the range as the measure of dispersion? (b) Which course has more dispersion in exam scores using the standard deviation as the measure of dispersion? (c) Suppose the score of 59.8 in the traditional course was incorrectly recorded as \(598 .\) How does this affect the range? the standard deviation? What property does this illustrate?

Find the population variance and standard deviation or the sample variance and standard deviation as indicated. $$ \text { Population: } 3,6,10,12,14 $$

For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning. (a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2011 (b) Most popular major for students enrolled in a statistics course (c) Average test score when the scores are distributed symmetrically (d) Average test score when the scores are skewed right (e) Average income of a player in the National Football League (f) Most requested song at a radio station (g) Typical number on the jersey of a player in the National Hockey League.

Mensa is an organization designed for people of high intelligence. One qualifies for Mensa if one's intelligence is measured at or above the 98 th percentile. Explain what this means.
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