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Chapter 2: Problem 121

Daily Calorie Consumption The five number summary for daily calorie consumption for the \(n=315\) participants in the NutritionStudy is \((445,\) 1334,1667,2106,6662) (a) Give the range and the IQR. (b) Which of the following numbers is most likely to be the mean of this dataset? Explain. $$ \begin{array}{llll} 1550 & 1667 & 1796 & 3605 \end{array} $$ (c) Which of the following numbers is most likely to be the standard deviation of this dataset? Explain. \(\begin{array}{lllll}5.72 & 158 & 680 & 1897 & 5315\end{array}\)

Short Answer

Expert verified
The range and IQR of the dataset are 6217 and 772, respectively. The most probable mean of the dataset is 1667. The most likely standard deviation of the dataset is 680.

Step by step solution

01

Range Calculation

The range of a dataset is calculated by subtracting the smallest number from the largest number. In this instance, it would be \( 6662 - 445 \)
02

IQR Calculation

The interquartile range (IQR) is calculated by subtracting the first quartile from the third quartile. Here, it should be \( 2106 - 1334 \)
03

Mean Identification

The mean of a dataset is its 'center' or 'balance point'. The value should be somewhere in the middle of the data range. Considering the choices given, the number that appears to be in the middle of our range is the median, 1667, so that's most likely to be our mean.
04

Standard Deviation Identification

The standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range. Considering the choices given and our calculated mean, the most suitable number is 680, which indicates a moderate level of variation.

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Key Concepts

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

Five-number summary
In descriptive statistics, a five-number summary provides a compact overview of a dataset. It helps you quickly grasp the distribution and spread of the data values by focusing on key points. The five numbers in this summary are:
  • Minimum: The smallest observation, which marks the lower extreme of the dataset.
  • First Quartile \((Q_1)\): Represents the 25th percentile, where 25% of data points are less than or equal to this value.
  • Median: The middle value, separating the lower and upper halves of the dataset.
  • Third Quartile \((Q_3)\): At the 75th percentile, indicating 75% of data is below this point.
  • Maximum: The largest observation, representing the upper extreme.
By looking at these five numbers, you get a sense of the dataset's range, central values, and the spread of the middle 50%. For the given exercise, the summary of daily calorie consumption values are: Minimum: 445, \((Q_1) = 1334\), Median = 1667, \((Q_3) = 2106\), and Maximum = 6662.
Interquartile range (IQR)
The interquartile range, or IQR, is a measure of statistical dispersion and is often used to understand the spread of the central 50% of a dataset. It is calculated by subtracting the first quartile (\(Q_1\)) from the third quartile (\(Q_3\)).The formula is: \[IQR = Q_3 - Q_1 \] For the dataset in the exercise, the IQR is \( 2106 - 1334 = 772\). This range between the quartiles illustrates the spread of the middle data points and gives insight into variability, offering a perspective that is resistant to extreme values (outliers). Analyzing the IQR helps in detecting data variability and is an essential part of statistical data analysis.
Standard deviation
Standard deviation is a key concept in statistics that reveals how dispersed or spread out the values in a dataset are. In simpler terms, it tells you how much the numbers differ from the average (mean) value. A smaller standard deviation suggests that the data points are generally close to the mean, while a larger one indicates greater variability. To determine which number is most likely the standard deviation of the dataset, you need a value that appropriately captures the spread of the calorie consumption data. It's important to note that:
  • Values close to the average mean less spread, indicating a more "compact" dataset.
  • Higher values suggest larger spread, indicating widely scattered data.
For the Nutrition Study data, a standard deviation of 680 fits because it shows a moderate spread around the mean. This reflects a balanced dispersion among the caloric intake of participants, capturing the true variance without exaggerating or underestimating it.

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Most popular questions from this chapter

Indicate whether the five number summary corresponds most likely to a distribution that is skewed to the left, skewed to the right, or symmetric. (100,110,115,160,220)

Making Friends Online A survey conducted in March 2015 asked 1060 teens to estimate, on average, the number of friends they had made online. While \(43 \%\) had not made any friends online, a small number of the teens had made many friends online. (a) Do you expect the distribution of number of friends made online to be symmetric, skewed to the right, or skewed to the left? (b) Two measures of center for this distribution are 1 friend and 5.3 friends. \({ }^{31}\) Which is most likely to be the mean and which is most likely to be the median? Explain your reasoning.

Donating Blood to Grandma? Can young blood help old brains? Several studies \(^{32}\) in mice indicate that it might. In the studies, old mice (equivalent to about a 70 -year-old person) were randomly assigned to receive blood plasma either from a young mouse (equivalent to about a 25 -year-old person) or another old mouse. The mice receiving the young blood showed multiple signs of a reversal of brain aging. One of the studies \(^{33}\) measured exercise endurance using maximum treadmill runtime in a 90 -minute window. The number of minutes of runtime are given in Table 2.17 for the 17 mice receiving plasma from young mice and the 13 mice receiving plasma from old mice. The data are also available in YoungBlood. $$ \begin{aligned} &\text { Table 2.17 Number of minutes on a treadmill }\\\ &\begin{array}{|l|lllllll|} \hline \text { Young } & 27 & 28 & 31 & 35 & 39 & 40 & 45 \\ & 46 & 55 & 56 & 59 & 68 & 76 & 90 \\ & 90 & 90 & 90 & & & & \\ \hline \text { Old } & 19 & 21 & 22 & 25 & 28 & 29 & 29 \\ & 31 & 36 & 42 & 50 & 51 & 68 & \\ \hline \end{array} \end{aligned} $$ (a) Calculate \(\bar{x}_{Y},\) the mean number of minutes on the treadmill for those mice receiving young blood. (b) Calculate \(\bar{x}_{O},\) the mean number of minutes on the treadmill for those mice receiving old blood. (c) To measure the effect size of the young blood, we are interested in the difference in means \(\bar{x}_{Y}-\bar{x}_{O} .\) What is this difference? Interpret the result in terms of minutes on a treadmill. (d) Does this data come from an experiment or an observational study? (e) If the difference is found to be significant, can we conclude that young blood increases exercise endurance in old mice? (Researchers are just beginning to start similar studies on humans.)

Rock-Paper-Scissors, also called Roshambo, is a popular two-player game often used to quickly determine a winner and loser. In the game, each player puts out a fist (rock), a flat hand (paper), or a hand with two fingers extended (scissors). In the game, rock beats scissors which beats paper which beats rock. The question is: Are the three options selected equally often by players? Knowing the relative frequencies with which the options are selected would give a player a significant advantage. A study \(^{10}\) observed 119 people playing Rock-Paper-Scissors. Their choices are shown in Table 2.6 . (a) What is the sample in this case? What is the population? What does the variable measure? (b) Construct a relative frequency table of the results. (c) If we assume that the sample relative frequencies from part (b) are similar for the entire population, which option should you play if you want the odds in your favor? (d) The same study determined that, in repeated plays, a player is more likely to repeat the option just picked than to switch to a different option. If your opponent just played paper, which option should you pick for the next round? $$\begin{array}{lc}\hline \text { Option Selected } & \text { Frequency } \\\\\hline \text { Rock } & 66 \\\\\text { Paper } & 39 \\\\\text { Scissors } & 14 \\\\\hline \text { Total } & 119 \\\\\hline\end{array}$$

Indicate whether the five number summary corresponds most likely to a distribution that is skewed to the left, skewed to the right, or symmetric. (15,25,30,35,45)
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