Waiting in Line The mean waiting time at the drive-through...

Waiting in Line The mean waiting time at the drive-through of a fast-food restaurant from the time an order is placed to the time the order is received is 84.3 seconds. A manager devises a new drivethrough system that he believes will decrease wait time. He initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table. $$ \begin{array}{rrrrr} \hline 108.5 & 67.4 & 58.0 & 75.9 & 65.1 \\ \hline 80.4 & 95.5 & 86.3 & 70.9 & 72.0 \\ \hline \end{array} $$ (a) Because the sample size is small, the manager must verify that wait time is normally distributed and the sample does not contain any outliers. The normal probability plot and boxplot are shown. The correlation between waiting time and expected $z$ -scores is 0.971 . Are the conditions for testing the hypothesis satisfied?

Waiting in Line The mean waiting time at the drive-through of a fast-food restaurant from the time an order is placed to the time the order is received is 84.3 seconds. A manager devises a new drivethrough system that he believes will decrease wait time. He initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table.
$$
\begin{array}{rrrrr}
\hline 108.5 & 67.4 & 58.0 & 75.9 & 65.1 \\
\hline 80.4 & 95.5 & 86.3 & 70.9 & 72.0 \\
\hline
\end{array}
$$
(a) Because the sample size is small, the manager must verify that wait time is normally distributed and the sample does not contain any outliers. The normal probability plot and boxplot are shown. The correlation between waiting time and expected $z$ -scores is 0.971 . Are the conditions for testing the hypothesis satisfied?

Key Concepts

Data Visualization for Assessing Distribution

Data visualization tools, such as normal probability plots and boxplots, are used to assess whether the underlying assumptions of a statistical test are met. A normal probability plot helps determine if the data follow a straight line, indicative of normality, while a boxplot can reveal the presence of outliers and provide a visual summary of the data's distribution.

Outlier Detection

Outlier detection involves identifying extreme values in the data that deviate markedly from other observations. Outliers can distort the results of statistical tests by affecting the mean and standard deviation. Ensuring that a dataset is free of significant outliers is critical for maintaining the validity of inference, particularly in small samples.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to support a particular belief about a population parameter. In the context of comparing a sample mean to a known value, a test such as the t-test is typically used to determine if the observed sample provides sufficient evidence to reject the null hypothesis, which usually assumes no change or effect.

Normality Assumption

The normality assumption is crucial when the sample size is small because many statistical tests, like the t-test, rely on the assumption that the data are drawn from a normally distributed population. This assumption helps ensure that the test statistic follows the expected distribution under the null hypothesis, which in turn validates inferences made from the test.

Small Sample Inference

In small sample inference, verifying assumptions such as normality becomes more important because the central limit theorem does not guarantee that the sample mean will be approximately normally distributed. This makes it essential to examine the actual distribution of the data in order to apply parametric tests reliably.

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