Question

{ Summarizing } & \text { Monthly } & \text { Retail } & \text { Sales } & \text { US }\end{array}\( monthly retail sales, in billions of dollars, for the 136 months starting with January 2000 is given in the RetailSales dataset, in the variable Sales, and shown in Figure \)2.29 .\( Use technology to find the mean and the standard deviation for this sample of US monthly retail sales. Use the \)95 \%\( Rule to compute an interval that is likely to contain about \)95 \%\( of the data. STATISTICS FOR NBA PLAYERS IN 20142015 Exercises 2.116 to 2.118 refer to the dataset NBAPlayers2015, which contains information on many variables for players in the NBA (National Basketball Association) during the \)2014-2015\( season. The dataset includes information for all players who averaged more than 24 minutes per game, and includes \)n=182$ players and 25 variables.

Short answer

For each dataset, the mean and standard deviation of the relevant variables need to be calculated first. Subsequently, the 95% Rule is applied to these results to construct intervals that are likely to contain about 95% of the data. The exact values would depend on the specific data in the datasets.

Step-by-step solution

Mean and Standard Deviation for Monthly Retail Sales

Load the RetailSales dataset into the statistical analysis tool of preference. Compute the mean and standard deviation of the 'Sales' variable, which represents the monthly retail sales. These operations are usually available in the descriptive statistics functions of most software.

Application of the 95% Rule to Retail Sales

The 95% Rule (or Empirical Rule) states that for a normal distribution, about 95% of the data will be within two standard deviations of the mean. Thus, construct an interval of the form: \( mean - 2*(standard deviation) \) to \( mean + 2*(standard deviation) \). This interval is likely to contain approximately 95% of the monthly retail sales data.

Mean and Standard Deviation for NBA Player's Data

Load the NBAPlayers2015 dataset into the statistical analysis tool. Compute the mean and standard deviation for the relevant variables of players who averaged more than 24 minutes per game. Similar to step 1, these operations can be done using the descriptive statistics functions of the software.

Application of the 95% Rule to NBA Player's Data

Similar to step 2, apply the 95% Rule for each variable by constructing intervals using the respective means and standard deviations. These intervals are likely to contain approximately 95% of the data for the individual variables.

Key concepts

Mean and Standard Deviation

Understanding the mean and standard deviation is essential in descriptive statistics, which involves summarizing and organizing data so that it can be easily understood. The mean, often referred to as the average, is calculated by adding all data points together and dividing by the number of points. It represents the central value of a data set.

Standard deviation, on the other hand, measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data are spread out over a large range of values.

For instance, in the context of US monthly retail sales, by calculating the mean, we get an idea of the typical sales amount. The standard deviation will then give us insight into the consistency of sales each month.

95% Rule

The 95% Rule is a part of the Empirical Rule, which is a statistical principle that applies to normal distributions. According to the 95% Rule, if the data follows a normal distribution, approximately 95% of the data should fall within two standard deviations of the mean—both above and below it.

In practical terms, you would calculate this interval by subtracting twice the standard deviation from the mean and adding twice the standard deviation to the mean. The resulting range is likely to contain about 95% of all the data points. For students and educators, this rule provides a quick way to understand the distribution of data and to estimate the spread of the bulk of data points.

Empirical Rule Application

Applying the Empirical Rule, which includes the 95% Rule, can help us understand the distribution of a dataset by providing a clear visual indication of spread. This is particularly useful in educational settings where students are first learning about data distribution.

The Empirical Rule requires the data to be normally distributed, which is an important assumption to check beforehand. Using tools to visualize the data, like histograms or box plots, can aid in this determination. Once the normal distribution is confirmed, you can use the Empirical Rule to predict probabilities for different ranges of data and to contextualize outliers.

Statistical Analysis

Statistical analysis encompasses a range of techniques used to make sense of data sets and to derive meaningful information out of them. In education, understanding statistical analysis can be a powerful tool for students to analyze research findings, interpret data in STEM subjects, and even to evaluate data in social sciences.

When analyzing datasets, like US monthly retail sales or the performance of NBA players, basic statistical operations like calculating the mean and standard deviation can reveal underlying patterns and variations in the data. More complex analyses might involve hypothesis testing, regression, or correlation analysis, which all depend on a solid understanding of the fundamental descriptive statistics.