Hygiene of handshakes, high fives, and fist bumps. Refer to...

Hygiene of handshakes, high fives, and fist bumps. Refer to the American Journal of Infection Control ( Aug. 2014) study of the hygiene of hand greetings, Exercise 9.24 (p. 479 ). The number of bacteria transferred from a gloved hand dipped into a culture of bacteria to a second gloved hand contacted by either a handshake, high five, or fist bump was recorded. Recall that the experiment was replicated only five times for each contact method and the data used to compare the mean percentage of bacteria transferred for any two contact methods. Therefore, for this independent-samples design, $n_{1}=n_{2}=5$. Suppose you want to estimate the difference between the mean percentage of bacteria transferred for the handshake and fist bump greetings to within $10 \%$ using a $95 \%$ confidence interval. a. Define the parameter of interest in this study. b. Give the value of $z_{\alpha / 2}$ for the confidence interval. c. What is the desired sampling error, SE? d. From the data provided in Exercise $9.24,$ find estimates of the variances, $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$, for the two contact methods. e. Use the information in parts a-d to calculate the number of replicates for each contact method required to obtain the desired reliability. Assume an equal number of replicates.

Hygiene of handshakes, high fives, and fist bumps. Refer to the American Journal of Infection Control ( Aug. 2014) study of the hygiene of hand greetings, Exercise 9.24 (p. 479 ). The number of bacteria transferred from a gloved hand dipped into a culture of bacteria to a second gloved hand contacted by either a handshake, high five, or fist bump was recorded. Recall that the experiment was replicated only five times for each contact method and the data used to compare the mean percentage of bacteria transferred for any two contact methods. Therefore, for this independent-samples design, $n_{1}=n_{2}=5$. Suppose you want to estimate the difference between the mean percentage of bacteria transferred for the handshake and fist bump greetings to within $10 \%$ using
a $95 \%$ confidence interval.
a. Define the parameter of interest in this study.
b. Give the value of $z_{\alpha / 2}$ for the confidence interval.
c. What is the desired sampling error, SE?
d. From the data provided in Exercise $9.24,$ find estimates of the variances, $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$, for the two contact methods.
e. Use the information in parts a-d to calculate the number of replicates for each contact method required to obtain the desired reliability. Assume an equal number of replicates.

Key Concepts

Variance Estimation

Variance estimation involves calculating or using sample estimates of the variability within each group to approximate the population variance. These estimates are crucial when comparing means from independent samples because the variability of the difference between means relies on the variances of both groups. Accurate estimation of variance directly impacts the reliability of the resulting confidence interval.

Sampling Error (Margin of Error)

In statistical estimation, the sampling error, or margin of error, represents the maximum expected difference between the true population parameter and the sample estimate. In constructing the confidence interval, it reflects how much the estimate might vary purely due to chance. In this study, the researcher desires the margin of error to be within 10% for the difference in bacterial transfer means.

Sample Size Calculation for Independent Samples

Sample size calculation is a fundamental concept in study design that determines how many replicates or observations are needed in each group to achieve a desired level of precision and reliability for the estimates. In the context of comparing two means with a pre-specified margin of error and confidence level, the calculation involves the z-critical value, the estimated variances, and the desired sampling error. This ensures that the experiment is adequately powered to detect the differences with the planned level of confidence.

Confidence Intervals

A confidence interval is a range of values, derived from the sample data, that is believed to cover the true population parameter with a specified probability. In this study context, a confidence interval is constructed around the difference between the two means and is intended to capture the true difference in bacterial transfer rates with 95% confidence.

Parameter of Interest

This concept refers to the specific numerical summary that the study is trying to estimate from the population. In the context of comparing two independent groups, it is the true difference between the population mean percentages of bacteria transferred by two different greeting methods. Rather than focusing on individual group means, the interest lies in how these means differ from each other in the population.

Critical Value from the z-distribution

The critical value, often denoted as z?/2 for a given level of significance, is a point on the standard normal distribution that corresponds to the desired level of confidence. For a 95% confidence interval, this value is typically approximately 1.96. This value is used to scale the standard error in order to construct the confidence interval.

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