One of your employees has suggested that your company develop a new product. You decide to take a random sample of your customers and ask whether or not there is interest in the new product. The response is on a 1 to 5 scale with 1 indicating "definitely would not purchase"; 2, "probably would not purchase"; 3, "not sure"; 4, "probably would purchase"; and 5, "definitely would purchase." For an initial analysis, you will record the responses 1, 2, and 3 as "No" and 4 and 5 as "Yes." What sample size would you use if you wanted the 90% margin of error to be 0.1 or less? (Round your answer up to the next whole number.)
________ participants
To determine the sample size required to achieve a 90% margin of error of 0.1 or less, we need to consider the following factors:
1. Confidence Level: The confidence level represents the probability that the sample accurately represents the population. In this case, we have a confidence level of 90%, which corresponds to a z-score of 1.645.
2. Margin of Error: The margin of error is the maximum amount of error we are willing to accept in our estimate. Here, we want it to be 0.1 or less.
3. Proportion of Interest: We want to estimate the proportion of customers who would potentially purchase the new product. Since we are considering responses 1, 2, and 3 as "No," and responses 4 and 5 as "Yes," we need to determine the proportion of customers who would respond with 4 or 5.
The formula to calculate the required sample size for a proportion estimation is:
n = (z^2 * p * (1 - p)) / (E^2)
Where:
- n is the required sample size,
- z is the z-score corresponding to the desired confidence level,
- p is the estimated proportion of customers interested in the new product,
- E is the maximum acceptable margin of error.
In this case, since we want to estimate the proportion of customers interested in the new product, we will use a conservative estimate of p = 0.5, assuming equal likelihood of customers being interested or not.
Plugging in the values, we have:
n = (1.645^2 * 0.5 * (1 - 0.5)) / (0.1^2)
n = (2.705 * 0.25) / 0.01
n ≈ 67.625
Since we cannot have fractional participants, we need to round up to the next whole number.
Therefore, you would need a sample size of 68 participants.