Problem 47 - Basic Statistics for Business & Economics

Question

It is estimated that \(60 \%\) of U.S. households subscribe to cable TV. You would like to verify this statement for your class in mass communications. If you want your estimate to be within 5 percentage points, with a \(95 \%\) level of confidence, how many households should you sample?

Step-by-Step Solution

Answer

You should sample 377 households.

Step 1: Identify the Proportion and Margins

We are given that the proportion of U.S. households with cable TV is 60%, which is represented mathematically as \( p = 0.60 \). We desire a margin of error of 5 percentage points, which means our margin of error \( E = 0.05 \).

Step 2: Formula for Sample Size

The formula to determine the sample size for a proportion estimate within a margin of error at a certain confidence level is \[ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right) \] where \( n \) is the sample size, \( Z \) is the Z-score corresponding to the desired confidence level, \( p \) is the proportion (0.60), and \( E \) is the margin of error (0.05).

Step 3: Determine the Z-score

For a 95% confidence level, the Z-score is approximately 1.96. This is the value derived from standard normal distribution tables.

Step 4: Plug Values into Formula

Substitute the values into the formula: \[ n = \left( \frac{1.96^2 \cdot 0.60 \cdot (1-0.60)}{0.05^2} \right) \].

Step 5: Calculate Numerator and Denominator

Calculate the numerator and denominator separately: - Numerator: \( 1.96^2 \cdot 0.60 \cdot 0.40 = 0.9408 \) - Denominator: \( 0.05^2 = 0.0025 \).

Step 6: Compute the Sample Size

Divide the numerator by the denominator to find \( n \): \[ n = \frac{0.9408}{0.0025} = 376.32 \].

Step 7: Round to the Nearest Whole Number

Since we cannot sample a fraction of a household, we round \( n = 376.32 \) up to the nearest whole number, which is 377.

Key Concepts

Confidence Interval

When working with data, we often want to estimate a population parameter. However, there's always a degree of uncertainty involved. The confidence interval provides a range around the estimate that is likely to contain the true population parameter with a specified level of confidence. For example, if we say the confidence interval is from 55% to 65% for cable TV subscription rates in U.S. households at a 95% confidence level, we mean we are 95% confident that the true subscription rate lies within this range. This concept allows decision-makers to understand the reliability of statistical estimations. It takes into account the natural variability in datasets to provide a more robust understanding.

Confidence levels commonly used are 90%, 95%, and 99%. The higher the confidence level, the wider the interval, showing increased certainty but also more range in the estimate.

Margin of Error

The margin of error quantifies the uncertainty in the estimate of a population parameter. It essentially tells us how far our estimate might be from the true population value. In our exercise, a margin of error of 5 percentage points means that the calculated cable TV subscription percentage (from our sample) might be 5 points higher or lower than the actual population parameter.

• The larger the sample size, the smaller the margin of error, assuming the confidence level remains constant.
• Conversely, a smaller sample size leads to a larger margin of error, implying less precise estimates.

This metric is crucial in survey designs, helping researchers determine both the necessary sample size and the reliability of the statistical results.

Z-score

The Z-score is a statistical measure that describes a value's position in relation to the mean of a group of values, measured in terms of standard deviations from the mean. In sample size determination, the Z-score is used to specify the desired confidence level. For a 95% confidence interval, the Z-score is typically 1.96. This value is derived from the standard normal distribution, indicating that the area under the curve between -1.96 and 1.96 captures 95% of the probability in a standard normal distribution.

• Higher Z-scores correspond to higher confidence levels and result in wider confidence intervals.
• Lower Z-scores are tied to lower confidence levels, resulting in narrower intervals.

Understanding Z-scores helps in assessing probabilities and making well-informed decisions based on statistical data.

Proportion Estimate

A proportion estimate is a guess about the portion of the population that exhibits a particular characteristic. In the context of our exercise, it's the percentage of U.S. households that subscribe to cable TV, initially estimated at 60%. Calculating a sample proportion helps verify or refute this assumption. The formula for determining the required sample size involves this estimated proportion, ensuring that the results are accurate and reliable.

When determining the sample size for a proportion, it's crucial to know:

• The expected proportion (from existing data or surveys).
• The required margin of error, which dictates how precise the estimate should be.
• The Z-score needed for the desired confidence level.

Using these elements helps refine decisions and validate real-world hypotheses, optimizing resources while still achieving definitive answers.