a veterinarians office wants to expand services the office surveyed a random sample of clients and asked which service they would most likely to be added the table shows the results

Question

grooming | boarding | day care | training lessons
16 | 9 | 21 | 19
The office has 195 total clients based on the results how many clients should the office expect to want day care services?

Answer 1

To determine how many clients should be expected to want day care services, you can use the results of the survey to calculate the proportion of clients who indicated they wanted day care, and then apply that proportion to the total number of clients.

1. Total Responses from the Survey: Sum the number of respondents for all services: \[ 16 \text{ (grooming)} + 9 \text{ (boarding)} + 21 \text{ (day care)} + 19 \text{ (training lessons)} = 65 \]

2. Proportion of Clients Wanting Day Care: The proportion of clients who want day care services is: \[ \text{Proportion for day care} = \frac{\text{number wanting day care}}{\text{total responses}} = \frac{21}{65} \]

3. Expected Clients Wanting Day Care Based on Total Clients: Now, to find out how many of the total 195 clients would want day care services, multiply the total number of clients by the proportion: \[ \text{Expected clients for day care} = 195 \times \frac{21}{65} \]

   Performing the calculation: \[ \text{Expected clients for day care} = 195 \times \frac{21}{65} = 195 \times 0.3231 \approx 63.83 \]

Since we can't have a fraction of a client, we can round to the nearest whole number. Therefore, the office should expect approximately 64 clients to want day care services.