Questions: A survey found that 16 out of 200 grocery store shoppers will donate 5 when asked by the cashier. a. What's the approximate probability that a random grocery store shopper will donate 5 when asked by the cashier? Type your answer as a percentage. % b. If 1000 people are asked to donate 5 by a cashier in one day, approximately how much money will be donated during that day?

Solution Steps

The probability \( p \) that a random grocery store shopper will donate \$5 when asked by the cashier is calculated as follows:

\[ p = \frac{\text{Number of donors}}{\text{Total shoppers}} = \frac{16}{200} = 0.08 \]

Expressed as a percentage, this probability is:

\[ p \times 100 = 8.00\% \]

Using the calculated probability \( p \), we can analyze the Bernoulli distribution. The probability of failure \( q \) is given by:

\[ q = 1 - p = 1 - 0.08 = 0.92 \]

The mean \( \mu \), variance \( \sigma^2 \), and standard deviation \( \sigma \) of the distribution are calculated as follows:

\[ \mu = p = 0.08 \]

\[ \sigma^2 = p \cdot q = 0.08 \cdot 0.92 = 0.0736 \]

\[ \sigma = \sqrt{pq} = \sqrt{0.08 \cdot 0.92} \approx 0.2713 \]

If 1000 people are asked to donate \$5, the expected number of donors is:

\[ \text{Expected donors} = 1000 \cdot p = 1000 \cdot 0.08 = 80 \]

The expected amount of money donated is then:

\[ \text{Expected donation} = \text{Expected donors} \cdot \text{Donation amount} = 80 \cdot 5 = 400.00 \]

Final Answer