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Chapter 5: Problem 54

Let \(x\) be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a. \(P(x=0)\) for \(n=5\) and \(p=.05\) b. \(P(x=4)\) for \(n=7\) and \(p=.90\) c. \(P(x=7)\) for \(n=10\) and \(p=.60\) Verify your answers by using Table I of Appendix \(\mathrm{C}\).

Short Answer

Expert verified
The solutions are found by plugging in the values into the binomial formula and carrying out the calculations. The results should then be cross-checked with those from Table I of Appendix C.

Step by step solution

01

Calculation using the binomial formula

The formula for calculating binomial probability is \(P(x; n, p) = \binom{n}{x} \cdot p^x \cdot (1 - p)^{n-x} \) where: \n\(\mathrm{n}\) is the number of trials,\n \(p\) is the probability of success in a single trial and\n \(x\) is the number of successful trials we're looking for. \n\n You will apply this formula in the questions using the given values for \(\mathrm{n, p, x}\).
02

Calculation for (a)

Use the formula to compute a.\nChoose \(\mathrm{n=5}\), \(\mathrm{p=.05}\), and \(\mathrm{x=0}\). \nCompute \(P(x=0;5.05) = \binom{5}{0} \cdot 0.05^0 \cdot (1-0.05)^5\).
03

Calculation for (b)

Use the formula to compute b.\nChoose \(\mathrm{n=7}\), \(\mathrm{p=.9}\), and \(\mathrm{x=4}\). \nCompute \(P(x=4;7,.9) = \binom{7}{4} \cdot 0.9^4 \cdot (1-0.9)^3\).
04

Calculation for (c)

Use the formula to compute c.\nChoose \(\mathrm{n=10}\), \(\mathrm{p=.6}\), and \(\mathrm{x=7}\). \nCompute \(P(x=7;10,.6) = \binom{10}{7} \cdot 0.6^7 \cdot (1-0.6)^3\).
05

Verification

Check with the statistical reference table I in Appendix C. \nFor each of the probability calculations, compare the result with the corresponding entry in the table. \nThe tabulated results might slightly differ from those calculated due to rounding errors or approximation.

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