Question

The weights (in pounds) of 27 packages of ground beef in a supermarket meat display are as follows: $$ \begin{array}{rrrrrrr} 1.08 & .99 & .97 & 1.18 & 1.41 & 1.28 & .83 \\ 1.06 & 1.14 & 1.38 & .75 & .96 & 1.08 & .87 \\ .89 & .89 & .96 & 1.12 & 1.12 & .93 & 1.24 \\ .89 & .98 & 1.14 & .92 & 1.18 & 1.17 & \end{array} $$ a. Draw a stem and leaf plot or a relative frequency histogram to display the weights. Is the distribution relatively mound-shaped? b. Find the mean and the standard deviation of the data set. c. Find the percentage of measurements in the intervals \(\bar{x} \pm s, \bar{x} \pm 2 s,\) and \(\bar{x} \pm 3 s\) d. How do the percentages in part c compare with those given by the Empirical Rule? Explain. e. How many of the packages weigh exactly 1 pound? Can you think of any reason for this?

Short answer

How many packages weigh exactly 1 pound, and what could be a possible reason for this observation?

Step-by-step solution

Determine Intervals

Calculate the intervals \(\bar{x} \pm s, \bar{x} \pm 2s,\) and \(\bar{x} \pm 3s\) using the mean and standard deviation found in the previous step.

Count Measurements in Intervals

Count the number of measurements or data points within each of the intervals calculated above.

Calculate Percentages

Divide the number of measurements within each interval by 27 (the total number of data points) and multiply by 100 to find the percentage of measurements in each of the intervals. #Fourth Part - Compare with Empirical Rule#

Review Empirical Rule

The Empirical Rule states that: - Approximately 68% of the data falls within \(\bar{x} \pm s\) - Approximately 95% of the data falls within \(\bar{x} \pm 2s\) - Approximately 99.7% of the data falls within \(\bar{x} \pm 3s\)

Compare Percentages

Compare the percentages calculated in the third part of the exercise to the percentages provided by the Empirical Rule. Discuss any differences observed. #Fifth Part - Packages Weighing Exactly 1 Pound and Reasoning#

Count Packages Weighing 1 Pound

Count the number of packages that weigh exactly 1 pound in the provided data.

Reasoning

Discuss any possible reasons for why there might be a particular number of packages weighing exactly 1 pound. This may involve considerations of consumer preferences, marketing strategies, or other factors.

Key concepts

Stem and Leaf Plot

A stem and leaf plot is a convenient method to display a small dataset, such as the weights of ground beef packages. This graphic organizer separates each data point into a stem (usually the leading digit or digits) and a leaf (usually the last digit). For instance, a weight of 1.08 pounds would be split into a stem of '10' and a leaf of '8'. To create the plot, one would list stems in a column and then write out each corresponding leaf next to its stem to display all the weights in the dataset in a visually intuitive manner.

When analyzing the stem and leaf plot, our goal is to determine the shape of the data distribution. A 'mound-shaped' or normal distribution typically appears symmetrical with most data points clustering around the center of the range and fewer points as we move away from the center. Drawing a stem and leaf plot can help us see if the data follows this pattern, which can then inform our analyses and expectations about the dataset.

Mean and Standard Deviation

Mean, commonly known as the average, is calculated by summing all data points in a dataset and dividing by the number of points. The mean represents the central location of the data. 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 are close to the mean, while a high standard deviation indicates that the data points are spread out over a wide range of values.

To find the mean and the standard deviation of the packages' weights, one combines all the weights, calculates the total, and divides by the total number of packages to get the mean. The standard deviation is then calculated using the formula that incorporates the mean and square differences from the mean for each data point. Both of these measures are crucial to statistical analysis, as they are used to summarize the center and spread of a dataset, respectively.

Empirical Rule

The Empirical Rule is extremely helpful in statistics as it offers a quick way to estimate the spread of data in a normal distribution. This rule tells us that for a normally distributed dataset:

• About 68% of the data falls within one standard deviation of the mean \( \bar{x} \pm s \).
• Approximately 95% lies within two standard deviations (\( \bar{x} \pm 2s \)).
• Roughly 99.7% should be found within three standard deviations (\( \bar{x} \pm 3s \)).

Upon calculating the percentage of weights within each interval around the mean, we can then compare these to the percentages predicted by the Empirical Rule. Discrepancies between our calculated values and the rule's predictions might indicate that our data does not follow a perfect normal distribution, or there may be other factors affecting our data. Understanding this principle is crucial when analyzing real-world data, as it provides context for the expected variability in measurements.