During the first quarter of \(2003,\) the price/earnings (P/E) ratio for stocks listed on the New York Stock Exchange generally ranged from 5 to 60 (The Wall Street Journal, March 7 , 2003 ). Assume that we want to estimate the population mean P/E ratio for all stocks listed on the exchange. How many stocks should be included in the sample if we want a margin of error of \(3 ?\) Use \(95 \%\) confidence.

When discussing statistics and sampling, the 'population mean' is a term that refers to the average value of a particular characteristic across the entire population. For example, in the context of the financial industry, if one wishes to understand the average Price-to-Earnings (P/E) ratio of all stocks listed on the New York Stock Exchange, the population mean would be the value to seek. Estimating the population mean with precision typically involves taking a randomly selected sample from the population and calculating the mean of that sample.

To ensure that this estimation is likely to reflect the population mean, an adequate sample size must be chosen. The population mean is a crucial starting point because any inaccuracy in its estimation will be reflected in subsequent statistical inferences or hypotheses tests about the population.

A 'confidence level' is a measure of certainty regarding the reliability of an estimate. It indicates the probability that the actual population parameter falls within a range, known as a confidence interval, around the estimated value. For instance, a 95% confidence level means that if 100 random samples were taken, the resulting confidence intervals from those samples would be expected to contain the actual population parameter 95 times out of 100. This confidence level is not about the likelihood of one sample being correct, but rather speaks to the long-term accuracy rate of the estimation process when repeated multiple times.

The choice of confidence level is dependent on how much risk of error one is willing to accept; higher confidence levels will result in a larger confidence interval, meaning a wider range of values around the estimated mean. In practice, common confidence levels include 90%, 95%, and 99% — with 95% being a standard choice in many fields, including the case of the P/E ratio estimation.

The 'margin of error' represents the extent to which an estimate obtained from a sample can differ from the true population parameter and still be considered accurate. Essentially, it quantifies the maximum expected difference between the sample estimate and the actual population value. The margin of error is influenced by two main factors: the level of confidence desired (with higher confidence necessitating a larger margin) and the variability within the population data. Narrower margins of error require larger sample sizes to maintain the same level of confidence.

In the P/E ratio estimation problem, a margin of error of 3 means we allow the sample mean to be within 3 units of the true population mean. A smaller margin of error, while desirable for more precise estimates, often necessitates a bigger sample which can be more costly or time-consuming to obtain.

In statistical analysis, the 'Z-score' is a standardized score that indicates the number of standard deviations a data point is from the population mean. Z-scores are used to calculate the probabilities of a score occurring within a normal distribution and are a key component in finding the correct sample size for estimating a population parameter with a specified confidence level and margin of error.

To determine the required sample size for our P/E ratio estimation with a 95% confidence level, we first identify the Z-score that corresponds to this level of confidence. For 95% confidence level, the Z-score is typically 1.96. This value is used in the sample size formula and reflects the number of standard deviations required to encapsulate 95% of the data in a normal distribution, ensuring that our estimated interval will likely contain the true population mean.

The 'normal distribution' is a continuous probability distribution that is symmetrical around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Also known as the Gaussian distribution, it is the foundational probability curve in statistics because of its widespread occurrences in natural and social phenomena.

When estimating the sample size for the P/E ratio problem, we assume that the distribution of P/E ratios is normally distributed, even though we don't know the actual distribution. This assumption allows us to use the standard normal distribution and its properties to determine the Z-score and subsequently the sample size. For a 95% confidence level, the Z-score tells us how many standard deviations away from the mean capture the central 95% of the data in a normal distribution, informing our calculations for estimate precision.