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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

As part of a semester project in a statistics course, Carlos surveyed a sample of 50 high school students and asked, "How many days in the past week have you consumed an alcoholic beverage?" The results of the survey are shown next. $$ \begin{array}{llllllllll} \hline 0 & 0 & 1 & 4 & 1 & 1 & 1 & 5 & 1 & 3 \\ \hline 0 & 1 & 0 & 1 & 0 & 4 & 0 & 1 & 0 & 1 \\ \hline 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ \hline 1 & 0 & 2 & 0 & 0 & 0 & 1 & 2 & 1 & 1 \\ \hline 2 & 0 & 1 & 0 & 1 & 3 & 1 & 1 & 0 & 3 \\ \hline \end{array} $$ (a) Is this data discrete or continuous? (b) Draw a histogram of the data and describe its shape. (c) Based on the shape of the histogram, do you expect the mean to be more than, equal to, or less than the median? (d) Determine the mean and the median. What does this tell you? (e) Determine the mode. (f) Do you believe that Carlos's survey suffers from sampling bias? Why?

The data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds. $$ \begin{array}{lll} \hline 2.0 & 1.9 & 1.8 \\ \hline 3.2 & 2.4 & 3.4 \\ \hline 1.6 & 2.7 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 8.4 & 7.2 & 7.6 \\ \hline 7.4 & 6.9 & 9.4 \\ \hline 9.1 & 8.1 & \\ \hline \end{array} $$ (a) Determine the mean and standard deviation of each data set. (b) Based only on the standard deviation, which data set has more spread? (c) What proportion of the observations is within one standard deviation of the mean for each data set? (d) The coefficient of variation, \(C V\), is defined as the ratio of the standard deviation to the mean of a data set, so $$ C V=\frac{\text { standard deviation }}{\text { mean }} $$ The coefficient of variation is unitless and allows for comparison in spread between two data sets by describing the amount of spread per unit mean. After all, larger numbers will likely have a larger standard deviation simply due to the size of the numbers. Compute the coefficient of variation for both data sets. Which data set do you believe has more "spread"? (e) Let's take this idea one step further. The following data represent the height of a random sample of 8 male college students. The data set on the left has their height measured in inches, and the data set on the right has their height measured in centimeters. $$ \begin{array}{lll} \hline 74 & 68 & 71 \\ \hline 66 & 72 & 69 \\ \hline 69 & 71 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 187.96 & 172.72 & 180.34 \\ \hline 167.64 & 182.88 & 175.26 \\ \hline 175.26 & 180.34 & \\ \hline \end{array} $$

The following data represent the age of U.S. presidents on their respective inauguration days (through Barack Obama). $$ \begin{array}{lllllllll} \hline 42 & 47 & 50 & 52 & 54 & 55 & 57 & 61 & 64 \\ \hline 43 & 48 & 51 & 52 & 54 & 56 & 57 & 61 & 65 \\ \hline 46 & 49 & 51 & 54 & 55 & 56 & 57 & 61 & 68 \\ \hline 46 & 49 & 51 & 54 & 55 & 56 & 58 & 62 & 69 \\ \hline 47 & 50 & 51 & 54 & 55 & 57 & 60 & 64 & \\ \hline \end{array} $$ (a) Find the five-number summary. (b) Construct a boxplot. (c) Comment on the shape of the distribution.

The following data represent the amount of time (in minutes) a random sample of eight students took to complete the online portion of an exam in Sullivan's Statistics course. Compute the range, sample variance, and sample standard deviation time. $$ 60.5,128.0,84.6,122.3,78.9,94.7,85.9,89.9 $$

Do store-brand chocolate chip cookies have fewer chips per cookie than Keebler's Chips Deluxe Chocolate Chip Cookies? To find out, a student randomly selected 21 cookies of each brand and counted the number of chips in the cookies. The results are shown next. $$ \begin{array}{lll|lll} &{\text { Keebler }} & && {\text { Store Brand }} \\ \hline 32 & 23 & 28 & 21 & 23 & 24 \\ \hline 28 & 28 & 29 & 24 & 25 & 27 \\ \hline 25 & 20 & 25 & 26 & 26 & 21 \\ \hline 22 & 21 & 24 & 18 & 16 & 24 \\ \hline 21 & 24 & 21 & 21 & 30 & 17 \\ \hline 26 & 28 & 24 & 23 & 28 & 31 \\ \hline 33 & 20 & 31 & 27 & 33 & 29 \\ \hline \end{array} $$ (a) Draw side-by-side boxplots for each brand of cookie. (b) Does there appear to be a difference in the number of chips per cookie? (c) Does one brand have a more consistent number of chips per cookie?
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