Problem 20 - A First Course in Probability

Question

If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100 people will contain (a) at least 50 who are in favor of the proposition; (b) between 60 and 70 inclusive who are in favor; (c) fewer than 75 in favor.

Step-by-Step Solution

Answer

The approximate probabilities for each scenario are as follows: (a) At least 50 people in favor: \(P(X ≥ 50) ≈ 0.9992\) (b) Between 60 and 70 people inclusive in favor: \(P(60≤X≤70) ≈ 0.7062\) (c) Fewer than 75 people in favor: \(P(X < 75) ≈ 0.9826\)

Step 1: Find the population proportion (p), and the standard deviation of the sample proportion

The given population proportion (p) is 65%, or 0.65. Since it is a binomial distribution, we can find the standard deviation of the sample proportion (σ) using the formula: \(σ = \sqrt{\frac{p(1-p)}{n}}\) where n is the sample size, which is 100 in this case. Plug in the numbers and we get: \(σ = \sqrt{\frac{0.65(1-0.65)}{100}} ≈ 0.0475\)

Step 2: Use the Z-score formula for each scenario

We will use the Z-score formula to find the probabilities for each scenario. The Z-score formula is: \(Z = \frac{\hat{p} - p}{σ}\) where \(\hat{p}\) is the sample proportion.

Step 3: Calculate the probability for scenario (a)

(a) At least 50 people are in favor of the proposition: To find the probability of at least 50 people in favor, we first find the Z-score for 50 people, or \(p_a = 0.50\). \(Z_a = \frac{0.50 - 0.65}{0.0475} ≈ -3.16\) Using a Z-table or a calculator, the probability for a Z-score of -3.16 is approximately 0.0008. Since we want the probability of at least 50 people in favor, we need to find the complement of this probability. \(P(X ≥ 50) = 1 - P(X<50) = 1 - 0.0008 = 0.9992\) The probability of a random sample of 100 people containing at least 50 in favor is approximately 0.9992.

Step 4: Calculate the probability for scenario (b)

(b) Between 60 and 70 people inclusive are in favor: To find the probability for this scenario, we need to find the Z-scores for 60 and 70 people, or \(p_b1 = 0.60\) and \(p_b2 = 0.70\). \(Z_{b1} = \frac{0.60 - 0.65}{0.0475} ≈ -1.05\) \(Z_{b2} = \frac{0.70 - 0.65}{0.0475} ≈ 1.05\) Using the Z-table or a calculator, the probabilities for Z-scores of -1.05 and 1.05 are approximately 0.1469 and 0.8531. The probability of a random sample of 100 people containing between 60 and 70 people in favor is: \(P(60≤X≤70) = P(Z_{b2}) - P(Z_{b1}) = 0.8531 - 0.1469 = 0.7062\)

Step 5: Calculate the probability for scenario (c)

(c) Fewer than 75 people are in favor: To find the probability of fewer than 75 people in favor, we first find the Z-score for 75 people, or \(p_c = 0.75\). \(Z_c = \frac{0.75 - 0.65}{0.0475} ≈ 2.11\) Using a Z-table or a calculator, the probability for a Z-score of 2.11 is approximately 0.9826. Since we want the probability of fewer than 75 people in favor, we don't need to find the complement of this probability. \(P(X < 75) = 0.9826\) The probability of a random sample of 100 people containing fewer than 75 in favor is approximately 0.9826.