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A specialty foods company sells "gourmet hams" by mail order. The hams vary in size from 4.15 to 7.45 pounds, with a mean weight of 6 pounds and standard deviation of 0.65 pounds. The quartiles and median weights are \(5.6,6.2,\) and 6.55 pounds. a. Find the range and the IQR of the weights. b. Do you think the distribution of the weights is symmetric or skewed? If skewed, which way? Why? c. If these weights were expressed in ounces \((1\) pound \(=16\) ounces \()\) what would the mean, standard deviation, quartiles, median, IQR, and range be? d. When the company ships these hams, the box and packing materials add 30 ounces. What are the mean, standard deviation, quartiles, median, IQR, and range of weights of boxes shipped (in ounces)? e. One customer made a special order of a 10 -pound ham. Which of the summary statistics of part d might not change if that data value were added to the distribution?

Step by step solution

01

Find the range and the IQR of the weights

The range of a set of data is the difference between the largest and smallest values. Here, it is \(7.45 - 4.15 = 3.3\) pounds. The IQR (Interquartile Range) is the difference between the first quartile (Q1) and the third quartile (Q3). So, the IQR is \(6.55 - 5.6 = 0.95\) pounds.

02

Analyze the distribution of the weights

To decide whether the distribution is symmetric or skewed, and if skewed, which way, we can look at the median and the mean. Since the mean (6 pounds) is less than the median (6.2 pounds), the distribution is skewed to the left. This is because in a negatively (left) skewed distribution, the mean is typically less than the median.

03

Convert weights to ounces

The weights can be expressed in ounces by multiplying by 16 (since \(1\) pound \(=16\) ounces). The mean becomes \(6 * 16 = 96\) ounces, the standard deviation becomes \(0.65 * 16 = 10.4\) ounces, the quartiles become \(5.6 * 16 = 89.6\) ounces, \(6.2 * 16 = 99.2\) ounces, and \(6.55 * 16 = 104.8\) ounces. The median becomes \(6.2 * 16 = 99.2\) ounces, the IQR becomes \(0.95 * 16 = 15.2\) ounces, and the range becomes \(3.3 * 16 = 52.8\) ounces.

04

Add the weight of the box and packing materials

When shipping the hams, the box and packing materials add 30 ounces to the weight. Add 30 to the mean, quartiles, median but not to the standard deviation, IQR, and range because these are measures of dispersion and are not affected by adding or subtracting constants. This gives a mean of \(96 + 30 = 126\) ounces, quartiles of \(89.6 + 30 = 119.6\) ounces, \(99.2 + 30 = 129.2\) ounces, and \(104.8 + 30 = 134.8\) ounces. The median becomes \(99.2 + 30 = 129.2\) ounces. The standard deviation, IQR, and range remain the same as in Step 3: \(10.4\) ounces, \(15.2\) ounces, and \(52.8\) ounces respectively.

05

Assess the impact of a 10-pound ham order on statistical measures

If a 10 -pound ham order (or 160 ounces when added the box and packing material becomes \(160 + 30 = 190\) ounces) is added to the distribution, the measures that would not change are the minimum, first quartile (Q1), median, and third quartile (Q3) because these are values that are determined by their position in an ordered data set and not affected by outliers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Range

The range in descriptive statistics is a way to measure the spread of your data set. It tells you the difference between the largest and smallest numbers in your set. In this case, the "gourmet hams" have a range in weight calculated by subtracting the smallest weight from the largest. This gives a range of 3.3 pounds (7.45 - 4.15 = 3.3). Understanding the range helps to provide an initial idea about the variability in the weights. It's important to note that while the range gives information about data spread, it can be sensitive to extremely high or low values called outliers.

Interquartile Range (IQR)

The interquartile range (IQR) is a measure of statistical dispersion and shows the spread of the middle 50% of your data. It is the difference between the third quartile (Q3) and the first quartile (Q1). For the "gourmet hams," with weights at Q1 as 5.6 pounds and Q3 as 6.55 pounds, the IQR is calculated as 0.95 pounds (6.55 - 5.6 = 0.95). This measurement is especially useful in identifying outliers because it focuses solely on the middle range and ignores extreme values.

Distribution Skewness

Skewness in statistics refers to the asymmetry in the distribution of data. If data is skewed, it means that data points are not symmetrically distributed around the mean. In the "gourmet hams" scenario, since the mean weight (6 pounds) is less than the median (6.2 pounds), this indicates a left or negative skewness. A negatively skewed distribution has a longer tail on the left side, suggesting that more data values fall to the right of the mean. Recognizing skewness helps in understanding the general shape of data distribution.

Measurement Conversion

Measurement conversion is fundamental in adjusting data from one unit to another, ensuring that comparisons or standardizations can be accurately made. Converting the weights from pounds to ounces—where 1 pound equals 16 ounces—incorporates simply multiplying each measurement by 16. For instance, the mean weight becomes 96 ounces (6 x 16). This conversion can be repeated for other statistical measures like standard deviation, quartiles, and IQR, allowing for the same analysis in a different unit of measurement.

Impact of Outliers

Outliers are data points significantly different from others in your dataset, and their presence can greatly affect statistical calculations. When one customer orders a significantly larger 10-pound ham, this value serves as an outlier. However, some measures like the quartiles and median remain unchanged, as they are determined by the order of data and robust against extreme values. Other statistical measures, such as the mean and range, could be influenced, though in this exercise, adding the 10-pound ham would mainly affect the mean and variance.