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Chapter 3: Problem 90

The weight-for-age for girls of 8 years is \(26 \mathrm{~kg}\) with a standard deviation of \(4.5 \mathrm{~kg}\). How heavy is an 8 -yearold girl with a z-score of \(-0.4 ?\) (Source: www.who.int)

Short Answer

Expert verified
The 8-year-old girl with a z-score of -0.4 weighs 24.2 kg.

Step by step solution

01

Understanding and arranging the formula

The formula for calculating z-score is Z = (X - 渭) / 蟽. We need to rearrange this formula to solve for X. This can be done by multiplying both sides by 蟽 and then adding 渭 to both sides. This gives us X = Z蟽 + 渭.
02

Substituting the given values into the formula

We are given that Z = -0.4, 蟽 = 4.5 kg and 渭 = 26 kg. We substitute these values into the formula we got from Step 1 to get X = (-0.4)(4.5) + 26.
03

Calculate the weight of the girl

Perform the necessary mathematical operations to find the value for X, which corresponds to the weight of the girl. Therefore, X = -1.8 + 26 = 24.2 kg

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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 that tells us how much the individual data points in a set generally vary from the average (or mean) of the set. In simpler terms, it tells us whether the numbers are spread out or clustered closely around the mean.
For example, in this exercise, the standard deviation is given as \(4.5\) kg, which means that the girl's weights vary roughly \(4.5\) kg from the average weight of \(26\) kg.
This value is critical as it helps us understand the distribution of data points in a dataset. A lower standard deviation means the data is more tightly packed around the mean, while a higher standard deviation indicates a spread-out dataset.
Normal Distribution
A normal distribution is a bell-shaped curve that is symmetrical around the mean. Many real-world phenomena, including heights, weights, and test scores, approximate a normal distribution. It's important because it allows us to predict probabilities and understand data patterns effectively.
In this case, the average weight of 8-year-old girls falls into this normal distribution category, with a mean (average) of \(26\) kg and a standard deviation of \(4.5\) kg. The curve symmetrically spreads around this average, informing us how typical or atypical a particular weight is for this age group. Normal distribution is crucial when utilizing z-scores, which help us determine how far a specific observation falls from the mean relative to the standard deviation.
Weight Calculation
Calculating the weight using z-score involves determining how far a data point is from the mean in terms of standard deviations. In this problem, we use the formula:
  • Z = (X - 渭) / 蟽
where \(Z\) is the z-score, \(X\) is the value we need to calculate, \(渭\) is the mean, and \(蟽\) is the standard deviation.
To solve for \(X\), rearrange the formula by multiplying both sides by \(蟽\) and adding \(渭\), resulting in \(X = Z蟽 + 渭\).
By substituting the given values: \(Z = -0.4\), \(蟽 = 4.5\), and \(渭 = 26\), we get:
  • \(X = (-0.4)(4.5) + 26\)
  • \(X = -1.8 + 26 = 24.2\) kg

This means the 8-year-old girl weighs \(24.2\) kg, indicating she is lighter than average by \(0.4\) standard deviations. Understanding weight calculation in the context of z-scores helps us make meaningful comparisons and analyses across datasets.

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Most popular questions from this chapter

The five-number summary for a distribution of final exam scores is $$ 60,78,80,90,100 $$ Is it possible to draw a boxplot based on this information? Why or why not?

The number of leaves taken per year by employees of an office has a mean of 50 and a standard deviation of 5 . a. What number of leaves corresponds to a \(z\) -score of \(1.25\) ? b. What number of leaves corresponds to a \(z\) -score of \(-1.75\) ?

The data that follow are final exam grades for two sections of statistics students at a community college. One class met twice a week relatively late in the day; the other class met four times a week at 11 a.m. Both classes had the same instructor and covered the same content. Is there evidence that the performances of the classes differed? Answer by making appropriate plots (including side-by-side boxplots) and reporting and comparing appropriate summary statistics. Explain why you chose the summary statistics that you used. Be sure to comment on the shape of the distributions, the center, and the spread, and be sure to mention any unusual features you observe. 11 a.m. grades: \(100,100,93,76,86,72.5,82,63,59.5,53,79.5\), \(67,48,42.5,39\) 5 p.m. grades: \(100,98,95,91.5,104.5,94,86,84.5,73,92.5,86.5\), \(73.5,87,72.5,82,68.5,64.5,90.75,66.5\)

In 2014 , the mean burglary rate (per 1000 houses) for all cities in a state was \(250 ;\) the standard deviation was 25 . Assume the distribution of burglary rates is approximately unimodal and symmetric. a. Approximately what percentage of cities would you expect to have burglary rates between 225 and 275 ? b. Approximately what percentage of cities would you expect to have burglary rates between 200 and 300 ? c. If someone guessed that the burglary rate in one of the cities was 0 , would you agree that that number was consistent with this data set?

This list represents the ages of the first six prime ministers of India when they first assumed office. (Source: www.pmindia.gov.in) $$ \begin{array}{ll} \text { Pandit Jawahar Lal Nehru } & 58 \\ \hline \text { Gulzari Lal Nanda } & 66 \\ \hline \text { Lal Bahadur Shastri } & 63 \\ \hline \text { Indira Gandhi } & 49 \\ \hline \text { Morarji Desai } & 81 \\ \hline \text { Charan Singh } & 72 \\ \hline \end{array} $$ a. Find the mean age, rounding to the nearest tenth. Interpret the mean in this context. b. According to a survey, people in the 20 th century had an average age of 80 years. How does the mean age of these prime ministers compare to that? c. Which of the prime ministers listed here had an age that is farthest from the mean and therefore contributes most to the standard deviation? d. Find the standard deviation, rounding to the nearest tenth.
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