From Allen (1990, 113, 162). The interactive computer system at Gnu Glue has 20 communication lines to the central computer system. The lines operate independently and the probability that any particular line is in use is 0.6. () What is the probability that 10 or more lines are in use? Compare the exact binomial solution and the normal approximation solution to this problem. (ii) Do the assumptions needed for the normal approximation make sense?

Binomial = 0.8724

Normal approximation = 0.873

Step-by-step explanation:

Given that :

Sample size (n) = 20

p = 0.6

q = 1 - p = 1 - 0.6 = 0.4

.using normal approximation :

Mean = np

Mean (m) = 20 * 0.6 = 12

Standard deviation (s) = √npq

s = √(20 * 0.6 * 0.4) = 2.1908902

Probability of 10 or more lines in use :

p(x ≥ 10)

Applying for correction:

p(x ≥ 10 - 0.5) = p(x ≥ 9.5)

Using the z formula :

Z = (x - m) / s

Z = (9.5 - 12) / 2.1908902

Z = - 1.141

p(Z > - 1.141) = 0.87307 (Z probability calculator)

p(Z > - 1.141) = 0.873

Using the binomial distribution :

P(x = x) = nCx * p^x * (1 - p)^(n-x)

p(x ≥ 10) = p(x = 10) + p(x = 11) + + p(x = 20)

Using calculator :

p(x ≥ 10) = 0.87247

= 0.873

2

step-by-step explanation:

1+1=2

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.