A bank with a branch located in a commercial district of a...

A bank with a branch located in a commercial district of a city has developed an improved process for serving customers during the noon-to-1 P.M. lunch period. The bank has the business objective of reducing the waiting time (defined as the time elapsed from when the customer enters the line until he or she reaches the teller window) to increase customer satisfaction. A random sample of 15 customers is selected (and stored in Bank1); the results (in minutes) are as follows: $$ \begin{array}{llllllll} 4.21 & 5.55 & 3.02 & 5.13 & 4.77 & 2.34 & 3.54 & 3.20 \\ 4.50 & 6.10 & 0.38 & 5.12 & 6.46 & 6.19 & 3.79 & \end{array} $$ Another branch, located in a residential area, is also concerned with the noon-to-1 P.M. lunch period. A random sample of 15 customers is selected (and stored in the file Bank2 ); the results (in minutes) are as follows: $$ \begin{array}{rrrrrrrr} 9.66 & 5.90 & 8.02 & 5.79 & 8.73 & 3.82 & 8.01 & 8.35 \\ 10.49 & 6.68 & 5.64 & 4.08 & 6.17 & 9.91 & 5.47 & \end{array} $$ a. Is there evidence of a difference in the median waiting time between the two branches? (Use $\alpha=0.05$.) b. What assumptions must you make in (a)? c. Compare the results (a) with those of Problem 10.12 (a) on page 376. Discuss.

A bank with a branch located in a commercial district of a city has developed an improved process for serving customers during the noon-to-1 P.M. lunch period. The bank has the business objective of reducing the waiting time (defined as the time elapsed from when the customer enters the line until he or she reaches the teller window) to increase customer satisfaction. A random sample of 15 customers is selected (and stored in Bank1); the results (in minutes) are as follows:
$$
\begin{array}{llllllll}
4.21 & 5.55 & 3.02 & 5.13 & 4.77 & 2.34 & 3.54 & 3.20 \\
4.50 & 6.10 & 0.38 & 5.12 & 6.46 & 6.19 & 3.79 &
\end{array}
$$
Another branch, located in a residential area, is also concerned with the noon-to-1 P.M. lunch period. A random sample of 15 customers is selected (and stored in the file Bank2 ); the results (in minutes) are as follows:
$$
\begin{array}{rrrrrrrr}
9.66 & 5.90 & 8.02 & 5.79 & 8.73 & 3.82 & 8.01 & 8.35 \\
10.49 & 6.68 & 5.64 & 4.08 & 6.17 & 9.91 & 5.47 &
\end{array}
$$
a. Is there evidence of a difference in the median waiting time between the two branches? (Use $\alpha=0.05$.)
b. What assumptions must you make in (a)?
c. Compare the results (a) with those of Problem 10.12 (a) on page 376. Discuss.

Key Concepts

Significance Level

The significance level, denoted as alpha (?), is the threshold for determining whether a test result is statistically significant. A significance level of 0.05 means that there is a 5% risk of concluding that a difference exists when there is none (Type I error). This level is commonly used in hypothesis testing to decide whether to reject the null hypothesis.

Comparison of Parametric and Nonparametric Methods

Comparing parametric methods like the t-test with nonparametric methods such as the Mann-Whitney U test allows for an evaluation of the robustness of the conclusions. Parametric methods assume specific distributions (often normal distribution) and focus on differences in means, while nonparametric methods focus on medians and are less sensitive to outliers and violations of normality. Differences in results may highlight the impact of distributional assumptions on the conclusions drawn from the data.

Assumptions in Nonparametric Testing

When using nonparametric tests for comparing medians, the primary assumptions typically include that the samples are independent, the response variable is measured at least on an ordinal scale, and that the distributions of the two populations have a similar shape. This ensures that differences detected are due to a shift in central tendency rather than differences in distribution shapes.

Hypothesis Testing

Hypothesis testing is a statistical technique used to decide whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. In this context, the test is used to determine whether there is a statistically significant difference in the central tendency (median) of waiting times between two independent samples.

Mann-Whitney U Test

The Mann-Whitney U test (also known as the Wilcoxon rank-sum test) is a nonparametric test used to compare the medians of two independent groups. It assesses whether the values in one population tend to be higher or lower than those in another by ranking all the observations from both groups together. This test is particularly appropriate when the assumption of normality is questionable or when the data are measured on an ordinal scale.

Nonparametric Hypothesis Testing

Nonparametric methods are statistical tests that do not require the assumption that the data follow a specific parametric distribution, such as the normal distribution. These tests, including the Mann-Whitney U test, are particularly useful when dealing with small sample sizes, ordinal data, or when the underlying distribution of the data is not known or is non-normal.

Median Comparison

The median is a measure of central tendency that represents the middle value in a data set when the values are arranged in order. Comparing medians is useful in situations where the data may be skewed or have outliers, as the median is less sensitive to extreme values than the mean.

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