Explain the reason that the expression, ∑ ( y ¯ i − y ¯ ) 2 ( g − 1 ) estimates the variance σ 2 n of the distribution of the means.

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Answer 1

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Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 14, Problem 67CP

a.

To determine

Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

b.

To determine

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

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Answer 2

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Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 14, Problem 67CP

a.

To determine

Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

b.

To determine

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

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Answer 3

Question

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 14, Problem 67CP

a.

To determine

Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

b.

To determine

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

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Answer 4

Question

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 14, Problem 67CP

a.

To determine

Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

b.

To determine

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

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Answer 5

Chapter 14, Problem 67CP

a.

To determine

Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

b.

To determine

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

Answer 6

Chapter 14, Problem 67CP

a.

To determine

Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

Answer 7

Chapter 14, Problem 67CP

a.

To determine

Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

Answer 8

b.

To determine

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

Answer 9

b.

To determine

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

Answer 10

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Answer 11

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Answer 12

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Answer 13

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Explain the reason that the expression, ∑ ( y ¯ i − y ¯ ) 2 ( g − 1 ) estimates the variance σ 2 n of the distribution of the means.

Solution Summary: The author explains the formula for sample variance, which is given by cVariance.

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Explain the reason that the expression, ∑(y¯i−y¯)2(g−1) estimates the variance σ2n of the distribution of the means.

Explain the reason that the expression, ∑n(y¯i−y¯)2(g−1) estimates σ2.

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