Questions: Time Magazine reports the "Divorce" rate has dropped to a 40-year low. Time's estimate is that 1.69% of married women get "Divorced" each year. We will take a sample of n=700 married women and model the proportion who got "Divorced" last year. (You will need to determine the mean and standard deviation for the p-hat model. Round your SD to four decimal places.) Question: P(25 or fewer women got "Divorced" last year) = Round your answer to four decimal places.

Solution Steps

To model the proportion of married women who got divorced, we use the binomial distribution with parameters:

• \( n = 700 \) (number of trials)
• \( p = 0.0169 \) (probability of success)

The mean \( \mu \) and standard deviation \( \sigma \) are calculated as follows:

\[ \mu = n \cdot p = 700 \cdot 0.0169 = 11.83 \]

\[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{700 \cdot 0.0169 \cdot (1 - 0.0169)} = \sqrt{11.6301} \approx 3.4103 \]

We need to find the cumulative probability \( P(X \leq 25) \), which is the sum of probabilities from \( P(X = 0) \) to \( P(X = 25) \).

Using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Calculating the probabilities for \( x = 0 \) to \( x = 25 \) yields:

• \( P(X = 0) = 0.0 \)
• \( P(X = 1) = 0.0001 \)
• \( P(X = 2) = 0.0005 \)
• \( P(X = 3) = 0.0019 \)
• \( P(X = 4) = 0.0057 \)
• \( P(X = 5) = 0.0136 \)
• \( P(X = 6) = 0.0272 \)
• \( P(X = 7) = 0.0463 \)
• \( P(X = 8) = 0.0689 \)
• \( P(X = 9) = 0.0911 \)
• \( P(X = 10) = 0.1083 \)
• \( P(X = 11) = 0.1167 \)
• \( P(X = 12) = 0.1152 \)
• \( P(X = 13) = 0.1048 \)
• \( P(X = 14) = 0.0884 \)
• \( P(X = 15) = 0.0695 \)
• \( P(X = 16) = 0.0512 \)
• \( P(X = 17) = 0.0354 \)
• \( P(X = 18) = 0.0231 \)
• \( P(X = 19) = 0.0142 \)
• \( P(X = 20) = 0.0083 \)
• \( P(X = 21) = 0.0046 \)
• \( P(X = 22) = 0.0025 \)
• \( P(X = 23) = 0.0012 \)
• \( P(X = 24) = 0.0006 \)
• \( P(X = 25) = 0.0003 \)

Summing these probabilities gives:

\[ P(X \leq 25) \approx 0.9996 \]

Final Answer