Question

Ethnic groups in America buy differing amounts of various food products because of their ethnic cuisine. A researcher interested in market segmentation for Asian and Hispanic households would like to estimate the proportion of households that select certain brands for various products. If the researcher wishes these estimates to be within. 03 with probability 95, how many households should she include in the samples? Assume that the sample sizes are equal.

Short answer

Answer: The required sample size for each group (Asian and Hispanic households) is 1068 households.

Step-by-step solution

Find the z-score for a 95% confidence level

The z-score corresponding to the middle 95% of a standard normal distribution is 1.96. This means that the area between -1.96 and 1.96 standard deviations around the mean contains 95% of the probability mass.

Use the formula to determine the required sample size

We will use the most conservative estimate for the proportion, which is 0.5. This is because choosing an extreme proportion (closer to 0 or 1) would result in a smaller required sample size in the formula, potentially underestimating the number of households needed for the study. Now, we will plug the values into the sample size formula: n = (Z^2 * p * (1-p)) / E^2 n = (1.96^2 * 0.5 * 0.5) / 0.03^2 Calculating the values, we get: n ≈ (3.8416 * 0.25) / 0.0009 n ≈ 0.9604 / 0.0009 n ≈ 1067.11 Since we cannot have a fraction of a household, we will round up to the nearest whole number: n ≈ 1068

Apply the sample size to both Asian and Hispanic households

The researcher should include 1068 households in each of the samples (Asian and Hispanic) to achieve the desired accuracy and confidence level. So, the researcher should include a total of 2136 households (1068 Asian and 1068 Hispanic) in the study.

Key concepts

Confidence Interval

When conducting a study, researchers often calculate a confidence interval to express the degree of uncertainty around an estimated population parameter. A confidence interval is a range of values, derived from the sample statistics, that likely contains the true population parameter. The '95% confidence level' mentioned in the original exercise indicates that if the same study were repeated multiple times, approximately 95% of the calculated intervals would contain the true population proportion.

In simpler terms, the confidence interval is like a net that we cast around our estimate, which we believe has a certain probability of catching the true value. The width of this net, the margin of error (E), influences how precise our estimate is. In the exercise, with E being 0.03, the net is cast tightly, seeking high precision in estimation of households' brand preferences for various products.

Z-score

A z-score is a statistical measure that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. It’s particularly useful in determining how unusual or typical a certain observation is within a dataset that follows a normal distribution.

The 1.96 z-score found in Step 1 of the solution is central to calculating the required sample size in the context of confidence intervals. Since the normal distribution is symmetrical, a z-score of 1.96 corresponds to the 97.5th percentile, meaning that 95% of the distribution falls between -1.96 and +1.96 z-scores. This standard score helps in determining the cushion needed around the estimated proportion to be confident that the true proportion falls within our confidence interval.

Population Proportion

The term population proportion refers to the percentage of a population that exhibits a particular characteristic. In our scenario, it represents the proportion of households within a certain ethnic group that prefer a particular brand. Estimating this proportion accurately is imperative for market segmentation and targeting.

In the calculation for the sample size in Step 2, the most conservative estimate (p = 0.5) was chosen because it maximizes the product of p*(1-p), resulting in the largest possible sample size. This conservative approach ensures that the sample size is sufficient regardless of the actual population proportion, which might not be known prior to conducting the research.

Market Segmentation

The concept of market segmentation involves dividing a broader market into subsets of consumers or households with common needs or interests, who are then targeted with specific products or marketing campaigns. By understanding the different preferences within each segment, companies can tailor their offers more effectively.

In terms of our exercise, knowledge of the proportion of Asian and Hispanic households that prefer certain brands allows for targeted marketing strategies in these segments. The careful determination of sample sizes ensures that the estimation of preferences within these groups will be accurate and reliable. Adequate sample sizes enable businesses to make informed decisions about product development, distribution, and promotional activities for different ethnic cuisines and preferences.