Problem 23 - Basic Statistics for Business & Economics

Question

A large on-demand video streaming company is designing a large-scale survey to determine the mean amount of time corporate executives watch on-demand television. A small pilot survey of 10 executives indicated that the mean time per week is 12 hours, with a standard deviation of 3 hours. The estimate of the mean viewing time should be within one-quarter hour. The \(95 \%\) level of confidence is to be used. How many executives should be surveyed?

Step-by-Step Solution

Answer

At least 554 executives should be surveyed.

Step 1: Identifying Parameters

The pilot survey gives us a sample mean of 12 hours and a sample standard deviation of 3 hours. We need to estimate the sample size that ensures the mean viewing time is within one-quarter of an hour with 95% confidence.

Step 2: Calculating the Margin of Error

The desired margin of error (E) is 0.25 hours because the estimate should be within one-quarter hour.

Step 3: Finding the Z-score for 95% Confidence

For a 95% confidence level, we use the standard normal distribution to find the Z-score, which is approximately 1.96.

Step 4: Using the Sample Size Formula

We use the formula for sample size estimation for a mean: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] Substitute the values: Z = 1.96, \(\sigma\) = 3, and E = 0.25.

Step 5: Calculating the Sample Size

Plug the values into the formula:\[ n = \left( \frac{1.96 \times 3}{0.25} \right)^2 = \left( \frac{5.88}{0.25} \right)^2 = (23.52)^2 = 553.6704 \]Since we cannot survey a fraction of a person, we round up to the next whole number, which is 554.

Key Concepts

Margin of Error

The margin of error is a crucial concept in statistics that helps define the accuracy of a survey or experiment's estimates. In simple terms, it tells us how far off our estimate could be from the true population parameter. In the exercise, the goal was to calculate how many executives need to be surveyed to ensure that the average viewing time is accurate within one-quarter hour, which is 0.25 hours. The formula linking sample size and margin of error is essential here:

• Lowering the margin of error requires a larger sample size.
• The margin of error decreases as the sample size increases, providing a more precise estimate.

Always remember, the smaller your margin of error, the more confident you can be that your sample accurately reflects the population.

Confidence Interval

A confidence interval gives a range of plausible values for a population parameter, such as a mean. In this exercise, the confidence interval determines how sure we need to be about our estimate. We specified a 95% level of confidence, meaning we want to be 95% sure that the true mean of executives' TV viewing time falls within our calculated interval. This is built upon:

• The sample mean as the center of the interval.
• The margin of error defining the width of the interval.

Larger confidence intervals offer greater confidence but are less precise, while smaller intervals are more precise but offer less confidence. For most surveys, a 95% confidence level is a standard choice, balancing confidence and precision.

Normal Distribution

Normal distribution, often known as the bell curve, is foundational to statistical analysis. It's a symmetrical distribution, with most data clustering around a central peak. The concept is essential when dealing with averages, as it underpins many statistical methods. In the context of sample size determination, the normal distribution allows us to utilize the Z-score method to find probabilities related to our margin of error and confidence interval. Key properties include:

• Symmetry about the mean.
• A relationship between standard deviation and data spread.
• Area under the curve representing probabilities.

The use of normal distribution simplifies calculations and enables estimations with known levels of accuracy and reliability.

Z-score

A Z-score, in the simplest terms, is a measure that describes a value's relationship to the mean of a group of values. By converting data into Z-scores, we can work out how far away data points are from the mean. In the survey planning exercise, the Z-score helps determine how many standard deviations a data point lies from the mean. For a 95% confidence level, the Z-score is approximately 1.96. Here's why Z-scores are powerful:

• They allow data from different scales to be compared easily.
• They play a key role in hypothesis testing and establishing confidence intervals.

Calculating Z-score is straightforward, helping statisticians in providing insights into data sets quickly and reliably.