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Chapter 1: Q52E (page 45)

The first four deviations from the mean in a sample of n=5 reaction times were .3, .9, 1.0, and 1.3. What is the fifth deviation from the mean? Give a sample for which these are the five deviations from the mean.

Short Answer

Expert verified

The fifth deviation from the mean is -3.5.

One of the samples is 5.3, 5.9, 6.0, 6.3, 1.5.

Step by step solution

01

Given information

The first four deviations from the mean were 0.3, 0.9, 1.0, 1.3.

The size of the sample is \(n = 5\).

02

Calculate the fifth deviation

Let x represents the fifth deviation.

We know that the sum of the deviations is 0.

Mathematically,

\(\begin{array}{c}0.3 + 0.9 + 1.0 + 1.3 + x &=& 0\\3.5 + x &=& 0\\x &=& - 3.5\end{array}\)

Thus, the fifth deviation is -3.5.

03

Obtain the sample of five observation

The sample observations are obtained as follows,

\({d_i} = {x_i} - \bar x \Rightarrow {{\bf{x}}_{\bf{i}}}{\bf{ = }}{{\bf{d}}_{\bf{i}}}{\bf{ + \bar x}}\)

Assuming the mean of the observations is 5\(\left( {\bar x} \right)\).\(\)

Thus, adding mean to each of the five deviations, the sample is given as,5.3, 5.9, 6.0, 6.3, 1.5.

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Most popular questions from this chapter

The National Health and Nutrition Examination Survey (NHANES) collects demographic, socioeconomic, dietary, and health related information on an annual basis. Here is a sample of \({\rm{20}}\) observations on HDL cholesterol level \({\rm{(mg/dl)}}\) obtained from the \({\rm{2009 - 2010}}\) survey (HDL is 鈥済ood鈥� cholesterol; the higher its value, the lower the risk for heart disease):

\(\begin{array}{l}{\rm{35 49 52 54 65 51 51}}\\{\rm{47 86 36 46 33 39 45}}\\{\rm{39 63 95 35 30 48}}\end{array}\)

a. Calculate a point estimate of the population mean HDL cholesterol level.

b. Making no assumptions about the shape of the population distribution, calculate a point estimate of the value that separates the largest \({\rm{50\% }}\) of HDL levels from the smallest \({\rm{50\% }}\).

c. Calculate a point estimate of the population standard deviation.

d. An HDL level of at least \({\rm{60}}\) is considered desirable as it corresponds to a significantly lower risk of heart disease. Making no assumptions about the shape of the population distribution, estimate the proportion \({\rm{p}}\) of the population having an HDL level of at least \({\rm{60}}\).

The article 鈥淓ffects of Short-Term Warming on Low and High Latitude Forest Ant Communities鈥� (Ecoshpere, May 2011, Article 62) described an experiment in which observations on various characteristics were made using mini chambers of three different types: (1) cooler (PVC

frames covered with shade cloth), (2) control (PVC frames only), and (3) warmer (PVC frames covered with plastic).One of the article鈥檚 authors kindly supplied the accompanying data on the difference between air and soil temperatures(掳C).

Cooler Control Warmer

1.59 1.92 2.57

1.43 2.00 2.60

1.88 2.19 1.93

1.26 1.12 1.58

1.91 1.78 2.30

1.86 1.84 0.84

1.90 2.45 2.65

1.57 2.03 0.12

1.79 1.52 2.74

1.72 0.53 2.53

2.41 1.90 2.13

2.34 2.86

0.83 2.31

1.34 1.91

1.76

a. Compare measures of center for the three different samples.

b. Calculate, interpret, and compare the standard deviations for the three different samples.

c. Do the fourth spreads for the three samples convey the same message as do the standard deviations about relative variability?

d. Construct a comparative boxplot (which was included in the cited article) and comment on any interesting features.

a. Give three different examples of concrete populations and three different examples of hypothetical populations.

b. For one each of your concrete and your hypothetical populations, give an example of a probability question and an example of an inferential statistics question.

The article cited in Example 1.2 also gave the accompanying strength observations for cylinders:

6.1

5.8

7.8

7.1

7.2

9.2

6.6

8.3

7.0

8.3

7.8

8.1

7.4

8.5

8.9

9.8

9.7

14.1

12.6

11.2


a. Construct a comparative stem-and-leaf display(see the previous exercise) of the beam and cylinder data, and then answer the questions in parts(b)鈥�(d) of Exercise 10 for the observations oncylinders.

b. In what ways are the two sides of the display similar? Are there any obvious differences between the beam observations and the cylinder observations?
c. Construct a dotplot of the cylinder data.

The following data on distilled alcohol content (%) for a sample of 35 port wines was extracted from the article 鈥淎 Method for the Estimation of Alcohol in Fortified Wines Using Hydrometer Baum茅 and Refractometer Brix鈥� (Amer. J. Enol. Vitic., 2006:486鈥�490). Each value is an average of two duplicate measurements.

16.35 18.85 16.20 17.75 19.58 17.73 22.75 23.78 23.25
19.08 19.62 19.20 20.05 17.85 19.17 19.48 20.00 19.97
17.48 17.15 19.07 19.90 18.68 18.82 19.03 19.45 19.37
19.20 18.00 19.60 19.33 21.22 19.50 15.30 22.25

Use methods from this chapter, including a boxplot that
shows outliers, to describe and summarize the data.
See all solutions

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