A survey taken by realtors in the San Francisco area found that 12 out of the 20 first-time home buyers sampled purchased their home with no-money- down loans. Calculate the probability that at least 12 in a sample of 20 first-time buyers would take out no-money-down loans if San Francisco's proportion is the same as the nationwide proportion of no- money-down loans.

#### Solution

Here $X$ denote the number of first time buyers who take out no-money-down loans. Assuming that San Francisco's proportion is the same as the nationwide proportion of no- money-down loans, `$p=12/20 = 0.6$`

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Given that `$X\sim B(20, 0.6)$`

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The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{20}{x} (0.6)^x (1-0.6)^{20-x},\\ &\quad x=0,1,\cdots, 20 \end{aligned} $$ `

The probability that at least 12 in a sample of 20 first-time buyers would take out no-money-down loans is `$P(X\geq 12)$`

.

` $$ \begin{aligned} P(X\geq 12) &= \sum_{x=12}^{20} P(x)\\ &= P(X=12) + P(X=13) + \cdots + P(X=20)\\ &= 0.1797+0.1659+0.1244\\ &\quad +0.0746+0.035+0.0123\\ &\quad +0.0031+0.0005+0\\ &= 0.5956 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators