Problem 27 - Basic Statistics for Business and Economics with Student

Question

A survey is being planned to determine the mean amount of time corporation executives watch television. A pilot survey indicated that the mean time per week is 12 hours, with a standard deviation of 3 hours. It is desired to estimate the mean viewing time within one quarter hour. The 95 percent level of confidence is to be used. How many executives should be surveyed?

Step-by-Step Solution

Answer

554 executives should be surveyed.

Step 1: Understand the Problem

To determine how many executives should be surveyed to estimate the mean viewing time within a given margin of error and confidence level, we have been given certain parameters: sample mean (12 hours), standard deviation (3 hours), margin of error (0.25 hours), and confidence level (95%).

Step 2: Identify the Formula for Sample Size

We will use the formula for sample size estimation for a mean: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]where \( n \) is the sample size, \( Z \) is the Z-score corresponding to the confidence level, \( \sigma \) is the standard deviation, and \( E \) is the margin of error.

Step 3: Find the Z-score for the Confidence Level

For a 95% confidence level, the Z-score is 1.96 because the 95% confidence interval corresponds to a Z-score of 1.96 in the standard normal distribution.

Step 4: Plug Values into the Formula

Substitute the given values into the formula: \( Z = 1.96 \), \( \sigma = 3 \), and \( E = 0.25 \).\[ n = \left( \frac{1.96 \times 3}{0.25} \right)^2 \]

Step 5: Calculate the Sample Size

First, calculate the expression inside the parentheses:\[ \frac{1.96 \times 3}{0.25} = \frac{5.88}{0.25} = 23.52 \]Then square the result:\[ n = (23.52)^2 \approx 553.5504 \]Since the sample size must be a whole number, we round up to the nearest whole number.

Step 6: Finalize the Sample Size

After rounding, the calculated sample size is 554. Thus, to achieve the desired margin of error with a 95% confidence level, 554 executives should be surveyed.

Key Concepts

Confidence Level

The confidence level in statistics tells us how certain or confident we can be about the results of a survey. It is usually expressed as a percentage. For example, if you have a confidence level of 95%, it means you can be 95% sure that the real mean (the average amount of time all executives watch TV) falls within the margin of error around your sample mean.
When calculating a confidence level, statisticians use the standard normal distribution. The Z-score associated with a 95% confidence level is 1.96, which corresponds to this interval. Essentially, this means that if you were to take 100 different samples, about 95 of them would have the true mean within the interval we have calculated.

Margin of Error

The margin of error is a statistical measurement that expresses the amount of random sampling error in a survey's results. It shows how much the survey results could differ from the true results if you were to survey the entire population.
In our exercise, we need the estimate of the mean viewing time to be accurate within one-quarter hour, which equals a margin of error of 0.25 hours. A smaller margin of error requires a larger sample size to ensure precision and confidence in the survey's findings. Thus, using this margin of error in our calculations, we determine how precise our estimate of the mean truly is.

Mean Estimation

Mean estimation involves determining the central value around which all other values in your data set congregate. It is colloquially referred to as the average. In the context of our exercise, mean estimation aims to find the average time executives spend watching TV.
The mean provided in the pilot survey tells us that, on average, executives watch television for 12 hours a week. Mean estimation is critical in helping businesses and researchers make informed decisions based on the average tendencies of a subject or phenomenon.

Standard Deviation

Standard deviation is a measure that assesses the dispersion or spread of data points in a data set. A low standard deviation means that most data points are close to the average, while a high standard deviation indicates a broader range of values.
In our scenario, the standard deviation was found to be 3 hours. This means that while some executives watch TV a little more or less than the average of 12 hours, most fall within 3 hours of this mean. Accurate knowledge of standard deviation helps in sizing the sample correctly, as it shows us the variability we might expect when estimating the mean value.