Determining Normality. In Exercises 9–12, refer to the indicated sample data and determine whether they appear to be from a population with a normal distribution. Assume that this requirement is loose in the sense that the population distribution need not be exactly normal, but it must be a distribution that is roughly bell-shaped.


Taxi Trips The distances (miles) traveled by New York City taxis transporting customers, as listed in Data Set 32 “Taxis” in Appendix B

Verified step by step guidance

1. Step 1: Visualize the data by creating a histogram or a boxplot of the distances traveled by the taxis. A roughly bell-shaped histogram or symmetric boxplot suggests normality.
2. Step 2: Calculate the descriptive statistics of the data, such as the mean, median, and standard deviation. For a normal distribution, the mean and median should be approximately equal.
3. Step 3: Perform a normal probability plot (also called a Q-Q plot). If the data points in the plot closely follow a straight line, this indicates that the data is approximately normally distributed.
4. Step 4: Conduct a formal statistical test for normality, such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test. These tests provide a p-value to assess whether the data significantly deviates from normality.
5. Step 5: Interpret the results. If the visualizations and statistical tests suggest that the data is roughly bell-shaped and does not significantly deviate from normality, you can conclude that the data appears to come from a population with a normal distribution.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is often represented as a bell-shaped curve, where the mean, median, and mode are all equal. Understanding this concept is crucial for determining if a dataset approximates a normal distribution.

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample means will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution. This theorem is fundamental in statistics because it justifies the use of normal distribution in inferential statistics, especially when dealing with large samples.

Normality Tests

Normality tests are statistical tests used to determine if a dataset follows a normal distribution. Common tests include the Shapiro-Wilk test and the Kolmogorov-Smirnov test. These tests provide a formal method to assess normality, which is essential for many statistical analyses that assume normality in the data.