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Chapter 7: Problem 33

The birth weights of full-term babies are normally distributed with mean \(\mu=3400\) grams and \(\sigma=505\) grams. Source: Based on data obtained from the National Vital Statistics Report, Vol. \(48,\) No. 3 (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of full-term babies who weigh more than 4410 grams. (c) Suppose the area under the normal curve to the right of \(x=4410\) is \(0.0228 .\) Provide two interpretations of this result.

Short Answer

Expert verified
Approximately 2.28% of full-term babies weigh more than 4410 grams.

Step by step solution

01

- Draw a normal curve

Draw a bell-shaped curve to represent a normal distribution. Label the center (mean) as \( \mu = 3400 \) grams, and mark points one, two, and three standard deviations away from the mean on either side (i.e., \( \mu \pm \sigma, \mu \pm 2\sigma, \mu \pm 3\sigma \)). Here, \( \sigma = 505 \) grams.
02

- Mark and shade the region

On the normal curve, locate the point \( x = 4410 \) grams. This value is \( 2\sigma \) away from the mean since \( 3400 + 2 \times 505 = 4410 \). Shade the area to the right of \( x = 4410 \), indicating the proportion of full-term babies who weigh more than 4410 grams.
03

- Interpret the area

The area to the right of \( x = 4410 \) is given as \ 0.0228 \. This can be interpreted as follows:
04

Interpretation 1

Approximately 2.28% of full-term babies weigh more than 4410 grams.
05

Interpretation 2

If you randomly select 100 full-term babies, about 2 or 3 of them will weigh more than 4410 grams.

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Key Concepts

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

Normal Distribution
Understanding the normal distribution helps you understand a wide variety of real-world scenarios, including birth weights. A normal distribution is often called a bell curve because of its shape. It is symmetric about the mean, which means it looks the same on the left and right sides. The curve is highest at the mean, and it falls off towards the tails.
  • Most data points lie close to the mean (center) of the curve.
  • Fewer data points lie farther from the mean, especially beyond 2 or 3 standard deviations.
This property helps in evaluating probabilities and natural variances in data, such as the birth weights mentioned in the problem.
Mean and Standard Deviation
The mean and standard deviation are essential components of the normal distribution.
  • The mean \( \mu = 3400 \) grams is the average birth weight of full-term babies.
  • The standard deviation \( \sigma = 505 \) grams tells you how spread out the weights are around the mean.
In simpler terms, the mean is where most birth weights are centered, and the standard deviation shows whether the weights are closely packed or widely spread out.
A smaller standard deviation means the weights are closer to the mean, while a larger standard deviation means they are more spread out.
Area Under the Curve
The area under the normal distribution curve represents probability or proportion. The total area under the curve is always equal to 1.
To solve the problem, you need to find the area to the right of \( x=4410 \) grams. This area (0.0228) represents the proportion of full-term babies who weigh more than 4410 grams.
  • Sketch the curve with the mean at the center (3400 grams), and mark 4410 grams.
  • Shade the area to the right of this point.
This shaded area (0.0228) tells you that 2.28% of full-term babies are expected to weigh more than 4410 grams.
Proportion Interpretation
Understanding proportions helps you relate abstract statistical concepts to real-world scenarios.
The given area of 0.0228 can be interpreted in two ways:
  • Approximately 2.28% of all full-term babies weigh more than 4410 grams.
  • If you randomly select 100 full-term babies, about 2 or 3 will weigh more than 4410 grams.
These interpretations convert the abstract area under the curve into concrete statements that are easier to grasp. They help you visualize how often such higher birth weights occur in reality.

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Most popular questions from this chapter

True or False: A normal score is the expected z-score of a data value, assuming the distribution of the random variable is normal.

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The ACT and SAT are two college entrance exams. The composite score on the ACT is approximately normally distributed with mean 21.1 and standard deviation 5.1 . The composite score on the SAT is approximately normally distributed with mean 1026 and standard deviation \(210 .\) Suppose you scored 26 on the \(\mathrm{ACT}\) and 1240 on the SAT. Which exam did you score better on? Justify your reasoning using the normal model.
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