According to the American Veterinary Medical Association (AVMA), 39.3% of households in the United States own a dog as a pet (AVMA website). Suppose that a company that sells dog food would like to establish a focus group to gather input on a new dog food marketing campaign. The company plans to contact 25 randomly selected households to invite people to join the focus group. For the sample of 25 households, compute standard deviation of households who own a dog as a pet. (to 4 decimals)

To solve this problem, we will use the formula for the standard deviation of a binomial distribution. Since each household either owns a dog or doesn't, this situation follows a binomial distribution, where the number of households with a dog can be modeled as a binomial random variable.

The formula for the standard deviation \( \sigma \) of a binomial distribution is given by:

\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\]

Where:
- \( n \) is the number of trials (households contacted) = 25
- \( p \) is the probability of success (households owning a dog) = 39.3% = 0.393
- \( 1 - p \) is the probability of failure (households not owning a dog)

Now, let's calculate it:

\[
\sigma = \sqrt{25 \cdot 0.393 \cdot (1 - 0.393)}
\]
\[
\sigma = \sqrt{25 \cdot 0.393 \cdot 0.607}
\]
\[
\sigma = \sqrt{25 \cdot 0.238251}
\]
\[
\sigma = \sqrt{5.956275}
\]
\[
\sigma \approx 2.4400
\]

Thus, the standard deviation of households who own a dog as a pet is approximately **2.4400**.

Would you like more details or have any further questions?

Here are five related questions for deeper understanding:
1. What is the expected number of households that own a dog in this sample?
2. How would increasing the sample size from 25 to 50 households affect the standard deviation?
3. What is the probability that exactly 10 households out of 25 own a dog?
4. How can we interpret the standard deviation in the context of this problem?
5. If the probability of dog ownership increased to 50%, how would the standard deviation change?

**Tip:** The binomial distribution is useful for modeling situations with two possible outcomes—like yes/no questions, successes/failures, or owning/not owning a dog.

Learn how to calculate the standard deviation of households owning a dog, based on the binomial distribution. Given that 39.3% of U.S. households own a dog, this problem computes the standard deviation for a sample of 25 randomly selected households. Ideal for high school statistics students.

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