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We examine the effect of \({ }^{37}\) Browne, M., et al., "Accumulation of Microplastic on Shorelines Worldwide: Sources and Sinks," Environmental Science and Technology, 2011; 45: 9175-9179. Data are approximated from information given. different inputs on determining the sample size needed. Find the sample size needed to give, with \(95 \%\) confidence, a margin of error within ±3 , if the estimated standard deviation is \(\tilde{\sigma}=100\). If the estimated standard deviation is \(\tilde{\sigma}=50\). If the estimated standard deviation is \(\tilde{\sigma}=10 .\) Comment on how the variability in the population influences the sample size needed to reach a desired level of accuracy.

Step by step solution

01

Calculation for Estimated Standard Deviation of 100

The Z-score associated with a 95% confidence level is 1.96. Substituting \[Z = 1.96\], \[\tilde{\sigma} = 100\], and \[E = 3\] into the formula, the sample size \[n = \left(\frac{1.96 * 100}{3}\right)^2 = 1073.7778\]. Since the sample size must be an integer, it is rounded up to 1074.

02

Calculation for Estimated Standard Deviation of 50

Using the same Z-score and margin of error, but changing the estimated standard deviation to 50, the sample size \[n = \left(\frac{1.96 * 50}{3}\right)^2 = 268.4444\]. Again, rounding this value gives a required sample size of 269.

03

Calculation for Estimated Standard Deviation of 10

Changing the estimated standard deviation to 10, the calculation is \[n = \left(\frac{1.96 * 10}{3}\right)^2 = 10.6889\], which rounds to yield a sample size of 11.

04

Discussion on Variability and Sample Size

The calculations demonstrate that as the estimated standard deviation (i.e., population variability) increases, the required sample size also increases. This is because when the population is more variable, a larger sample is needed to accurately estimate the population parameter with the desired level of confidence and margin of error.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Level

Confidence level represents how certain we are that our sample's estimate will fall within the margin of error. Imagine a survey predicting an election's outcome. If you say you are 95% confident in the results, you're effectively saying that if we were to repeat the survey multiple times, 95% of those surveys would reflect the true election result within the specified margin of error.

In statistics, the confidence level is depicted as a percentage, most common being 90%, 95%, and 99%. Higher confidence levels provide more certainty but may require a larger sample size. This is why many studies prefer using a standard of 95% confidence, which balances accuracy and practicality.

• Choosing a higher confidence level means greater certainty that your sample reflects the true population.
• However, it often demands a larger sample size to achieve the same level of precision (margin of error).
• The Z-score in statistical terms relates directly to confidence levels; for instance, a 95% confidence level corresponds to a Z-score of 1.96.

Margin of Error

Margin of error is a critical concept in determining how precise your estimate is. It is a measure of the range within which the true population parameter is expected to lie, based on your sample statistic.

For example, if a survey reports that 60% of respondents support a policy with a margin of error of ±3%, it means that if you surveyed the entire population, the actual percentage would likely be between 57% and 63%.

• Often presented as a plus-minus range, it balances precision with practicality.
• A smaller margin of error requires a larger sample size but gives more reliable results.
• The desired margin of error directly influences the calculation of sample size in your study.
• Reducing the margin of error by half may require increasing the sample size by four times, under common statistical conditions.

Standard Deviation

Standard deviation is an important measure of the amount of variation or dispersion in a set of values. In the context of sample size determination, it reflects the variability within the population that we want to estimate.

A larger standard deviation signifies that the data points are spread out over a broader range of values, whereas a smaller standard deviation indicates that the data points are closer to the mean.

• The sample size needed for statistical significance increases with greater population variability.
• If the estimated standard deviation is high, it suggests more spread in data, thus needing a larger sample for accuracy.
• For instance, using a smaller standard deviation, such as 10 compared to 100, greatly decreases the sample size needed to maintain the same confidence and margin of error.
• This is shown in the calculations where a larger standard deviation required a greater sample size to achieve the same level of precision and confidence.