Online Gambling Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be 99% confident that your estimate is in error by no more than two percentage points?

b. Assume that 18% of all adults gamble online (based on 2017 data from a Gambling Commission study in Great Britain).

Verified step by step guidance

Step 1: Identify the key components of the problem. The confidence level is 99%, the margin of error (E) is 2 percentage points (0.02), and the estimated proportion (p̂) is 18% (0.18). The formula for determining the required sample size is: n = (Z² * p̂ * (1 - p̂)) / E².

Step 2: Determine the Z-value corresponding to a 99% confidence level. For a 99% confidence level, the Z-value is approximately 2.576. This value is derived from the standard normal distribution table.

Step 3: Substitute the values into the formula. Using the formula n = (Z² * p̂ * (1 - p̂)) / E², substitute Z = 2.576, p̂ = 0.18, and E = 0.02. The formula becomes: n = ((2.576)² * 0.18 * (1 - 0.18)) / (0.02)².

Step 4: Simplify the expression inside the formula. First, calculate (1 - p̂), which is (1 - 0.18) = 0.82. Then calculate the numerator: (2.576)² * 0.18 * 0.82. Finally, calculate the denominator: (0.02)².

Step 5: Divide the numerator by the denominator to find the required sample size. Round up the result to the nearest whole number, as the sample size must be an integer.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Determination

Sample size determination is a statistical method used to calculate the number of observations or replicates needed in a survey to achieve a desired level of confidence and precision. In this context, it involves using the margin of error, confidence level, and population proportion to find the minimum number of adults to survey to ensure that the estimate of online gambling participation is accurate within two percentage points.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the confidence interval calculated from the sample data. A 99% confidence level indicates that if the same survey were conducted multiple times, 99% of the calculated intervals would contain the true proportion of adults who gamble online. This high level of confidence is crucial for making reliable marketing decisions.

Margin of Error

The margin of error quantifies the range within which the true population parameter is expected to fall, based on the sample data. In this scenario, a margin of error of two percentage points means that the estimate of the percentage of adults who gamble online could be two points higher or lower than the calculated sample proportion. This concept is essential for understanding the precision of the survey results.

Watch next

Master Constructing Confidence Intervals for Proportions with a bite sized video explanation from Patrick

Start learning