Question

In Exercise 6.93 on page \(430,\) we see that the average number of close confidants in a random sample of 2006 US adults is 2.2 with a standard deviation of \(1.4 .\) If we want to estimate the number of close confidants with a margin of error within ±0.05 and with \(99 \%\) confidence, how large a sample is needed?

Short answer

After evaluating and rounding up, you will find you need a sample size of approximately 1237 people to obtain an estimate within the ±0.05 margin of error with a \(99\%\) confidence level.

Step-by-step solution

Understand the Given Variables

The exercise provides the standard deviation (\(1.4\)), a desired margin of error (\(0.05\)), and a confidence level of \(99\% .\) Note that for a \(99\%\) confidence level, the Z-score (which can be found on the Z-table or standard normal distribution table) is \(2.576\).

Apply the Confidence Interval Formula

The formula we'll use to solve the exercise is the one for the sample size needed for a confidence interval, which is given as: \n Sample size \(n = \left( \frac{Z_{\frac{\alpha}{2}} \cdot \sigma}{E} \right)^2 \). In this formula, \(Z_{\frac{\alpha}{2}}\) is the Z-score, \(\sigma\) is the standard deviation and \(E\) is the margin of error.

Plug in the Given Values

We can substitute the provided data into the formula: \(n = \left( \frac{2.576 \cdot 1.4}{0.05} \right)^2\). This needs to be evaluated and the resulting number should be rounded up to the next whole number, because we can't work with fractions of a person.

Key concepts

Confidence Interval

When researchers conduct studies, they're often interested in understanding a wider population based on a smaller group, or sample, taken from it. To estimate a population parameter like the average number of close confidants among US adults, instead of providing a single number, they give a range of numbers that they believe the actual average is likely to fall within. This range is known as a confidence interval, which gives both the estimated value and a sense of the precision of the estimate. The width of this interval depends on the desired confidence level and variability in the data. A 99% confidence interval is especially wide, implying that there's a high degree of certainty that the population parameter lies within that range. However, to maintain a small margin of error with such a high confidence level typically requires a larger sample size.

Margin of Error

The margin of error reflects the extent of uncertainty involved in the sample estimate. It is the maximum expected difference between the true population parameter and a sample estimate of that parameter. In other words, it tells us how close we can expect our sample statistic to be to the population value. A smaller margin of error indicates greater precision in the sample estimate but usually necessitates a larger sample size to achieve that precision. For example, being within ±0.05 of the true average number of close confidants means researchers can be quite certain that their sample provides an accurate estimate of the actual number, but as a trade-off, they’ll need to collect data from more individuals compared to a larger margin of error.

Standard Deviation

Variability in data is quantified using a measure called the standard deviation. It indicates how much individuals within a dataset deviate from the average or mean value. In the context of sample size determination, the standard deviation plays a crucial role as it affects the width of the confidence interval. A high standard deviation signifies that data points are spread out widely from the mean, potentially increasing the required sample size to achieve a specific margin of error. Conversely, a low standard deviation means data points are clustered closely around the mean, which may allow a smaller sample to still accurately estimate the population parameter with a given level of confidence. The known standard deviation of 1.4 in the close confidants example informs us about the variability in the number of confidants among US adults.

Z-score

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. For sample size determination, specifically when it's associated with a confidence interval, the Z-score corresponds to the desired confidence level. It tells us how many standard deviations an element is from the mean. To find the required Z-score for a specific confidence level, such as our 99%, statisticians typically use a Z-table. The Z-score of 2.576, for instance, indicates that the desired confidence level will include the mean plus or minus 2.576 standard deviations. This Z-score helps to scale the margin of error according to the level of confidence we wish to achieve, which in turn influences the calculation of the required sample size.