# Calculating Sample Sizes

Throughout this week’s lectures in our Marketing Research & Strategy class, we’ve been taking a closer look at how to determine a sample size for a research project. McQuarrie (2016) provides a relatively easy way to calculate how big your sample size should be in his book The Market Research Toolbox: A Concise Guide for Beginners. In his book, he describes a three-step process that will help you calculate the sample size based on the preferred confidence level and margin of error (precision) that is proposed by management judgement:

1. Square the Z value associated with the desired confidence interval.
2. Multiply it by the population variance.
3. Divide by the square of the desired precision.

To find the population variance, you have to use the following formula:

Variance = proportion #1 x [1 – proportion #1]

Now we know how to theoretically calculate the sample size, we can apply this to a problem. One of the problems that was presented by McQuarrie (2016) stated: “To determine the effectiveness of an ad campaign for a new DVD player, management would like to know what percentage of the market has been made aware of the new product. The ad agency thinks this figure could be as high as 70 percent. In estimating the percent aware, management has specified a 95 percent confidence interval, and a precision of ±2 percent. What sample size is needed?”

Following the method presented by McQuarrie for calculating the sample size, the first thing we need to do is to square the Z value associated with the confidence interval. The problem states that management decided on a confidence interval of 95 percent which means that our Z value equals 2. In the next step, we have to multiply our squared Z value with the population variance, which can be calculated through the formula shown above for variance. In this case the variance equals 0.21 [0.70 x (1-0.70)]. Once, we’ve established this, we have to divide our nominator (Z2 x variance) through our denominator, which equals the square of the desired precision. This means that our final formula will look like this:

[22 x 0.21] / 0.022
= 2100

I think a margin of error (precision) of ±2 percent is a reasonable confidence interval. At first, it seemed really tight but after taking a closer look at an article by Billy Hulkower, a Senior Technology Analyst for Mintel, on the market for Movie Sales and Rentals in the US in 2014, we can conclude that the market for movie sales and rentals is declining rapidly. The tables show that movie sales in the US, when adjusted for inflation will decline from \$17.5 billion in 2014 to \$14.6 in 2019. Based on this information, it is extremely important for a company, that is about to introduce a new DVD player into the market, to know how effective their ad campaign will be.

Another reason why it is so important for companies in the DVD player market to know how effective their ad campaign will be, is the increasing competition of digital movies provided by for example Amazon Instant Video, iTunes, and Google Play which is also discussed in Hulkower’s article. Technology is constantly evolving and helps us make our lives easier. Customers now have the option between buying movies at home from their computers or running to the store to physically buy the movie. I think we can all agree that it is a lot more attractive to stay at home and buy a movie online without having to leave your couch instead of driving all the way to the store for that same movie.

Due to these two reasons, a ±2 percent precision level in this problem seems a very reasonable estimate, because as a company in a declining market with a lot of competition wants to get an accurate reflection of the percentage of the market that is aware of the new DVD player that you are to introduce into the market.