Explanation Definition: 1. While computing the mean of a frequency distribution, write all products x ⋅ f of the values and their frequencies in a new column of the table. Represent the sum of these products by ∑ ( x ⋅ f ) . Denote the sum of the frequencies by ∑ f . The mean is then ∑ ( x ⋅ f ) ∑ f . 2. If x is a data value in a set whose mean is x ¯ , then x − x ¯ is called x’s deviation from the mean. 3. We denote the standard deviation of a sample of n data values by s = ∑ ( x − x ¯ ) 2 n − 1 . 4. We calculate the standard deviation, s, of a sample that is given as a frequency distribution as s = ∑ ( x − x ¯ ) 2 ⋅ f n − 1 . Here f is the frequency of data value x , and n = ∑ f is the number of data values in the distribution. Calculation: Observe the figure carefully. Notice that the grades depend upon the mean and the standard deviation. To determine the grade we will first determine the mean and standard deviation. From the given diagram understand the frequency of each data value. Average value x Frequency f 80 2 81 5 82 2 First find the mean of the given data set. To compute the mean, let us find all products x ⋅ f of the given values and their respective frequencies. For x = 80 and f = 2 , the product x ⋅ f = 80 ⋅ 2 = 160. For x = 81 and f = 5 , the product x ⋅ f = 81 ⋅ 5 = 405. For x = 82 and f = 2 , the product x ⋅ f = 82 ⋅ 2 = 164. The sum of these products is 160 + 405 + 164 = 729. Therefore, ∑ ( x ⋅ f ) = 729. The sum of the frequencies is 2 + 5 + 2 = 9. Therefore, ∑ f = 9

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Chapter 14.3, Problem 30E

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The number of Fs assigned using the given diagram, if the averages in the class are 80, 81, 81, 82, 81, 81, 82, 80, and 81

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Chapter 14.3, Problem 29E

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Q1) Classify the following statements as a true or false statements a. Any ring with identity is a finitely generated right R module.- b. An ideal 22 is small ideal in Z c. A nontrivial direct summand of a module cannot be large or small submodule d. The sum of a finite family of small submodules of a module M is small in M A module M 0 is called directly indecomposable if and only if 0 and M are the only direct summands of M f. A monomorphism a: M-N is said to split if and only if Ker(a) is a direct- summand in M & Z₂ contains no minimal submodules h. Qz is a finitely generated module i. Every divisible Z-module is injective j. Every free module is a projective module Q4) Give an example and explain your claim in each case a) A module M which has two composition senes 7 b) A free subset of a modale c) A free module 24 d) A module contains a direct summand submodule 7, e) A short exact sequence of modules 74.

************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.

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The number of Fs assigned using the given diagram, if the averages in the class are 80, 81, 81, 82, 81, 81, 82, 80, and 81

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To find: The number of Fs assigned using the given diagram, if the averages in the class are 80, 81, 81, 82, 81, 81, 82, 80, and 81

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To find:

The number of Fs assigned using the given diagram, if the averages in the class are 80, 81, 81, 82, 81, 81, 82, 80, and 81