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Chapter 2: Problem 8

An article in Consumer EX0208 Reports gives the price-an estimated average for a 6-ounce can or a 7.06 -ounce pouch \(-\) for 14 different brands of water- packed light tuna, based on prices paid nationally in supermarkets: $$ \begin{array}{rrrrrrr} .99 & 1.92 & 1.23 & .85 & .65 & .53 & 1.41 \\ 1.12 & .63 & .67 & .69 & .60 & .60 & .66 \end{array} $$ a. Find the average price for the 14 different brands of tuna. b. Find the median price for the 14 different brands of tuna. c. Based on your findings in parts a and b, do you think that the distribution of prices is skewed? Explain.

Short Answer

Expert verified
Answer: The mean price for the 14 different brands of tuna is $0.74, and the median price is $0.665. Based on these values alone, it is difficult to strongly conclude if the price distribution is skewed.

Step by step solution

01

Compute the mean price.

First, add up all the prices and then divide the sum by the total number of data points (14) to get the mean value for the given data set. Mean = \(\frac{\sum_{i=1}^{14}x_i}{14}\) Sum up the prices: \(.99 + 1.92 + 1.23 + .85 + .65 + .53 + 1.41 + 1.12 + .63 + .67 + .69 + .60 + .60 + .66\) Sum = \(10.35\) Mean = \(\frac{10.35}{14} = 0.74\) So, the average price for the 14 different brands of tuna is $0.74.
02

Compute the median price.

First, arrange the data set in ascending order. Then, since we have an even number of data points, the median is the average of the two middle values. The sorted data set: .53, .60, .60, .63, .65, .66, .67, .69, .85, .99, 1.12, 1.23, 1.41, 1.92 Middle values: .66 and .67 Median = \(\frac{(.66 + .67)}{2}=0.665\) So, the median price for the 14 different brands of tuna is $0.665.
03

Determine if the price distribution is skewed.

By comparing the mean (\(0.74) and the median (\)0.665), we notice that the mean is greater than the median. This could indicate a positive/right-skewed distribution. However, as the difference is not significant, we cannot be certain. Based on these values alone, it might be hard to definitively claim that the distribution of prices is skewed. A histogram or further analysis might be necessary to make a stronger conclusion about the skewness of the price data. In conclusion, we have found the mean and median prices for the 14 different brands of water-packed light tuna as \(0.74 and \)0.665, respectively. From the given data, it is difficult to make a strong conclusion about the skewness of the price distribution.

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Key Concepts

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

Mean Calculation
Understanding the mean, or average, of a data set is a fundamental skill in both math and real-world applications. It is the sum of all the values divided by the number of values. In the case of the tuna prices, we first sum all the 14 prices to get the total cost. After that, we divide by 14 to find the average.

For instance, if you were to buy one can of each brand, the mean price would tell you the average cost per can, giving you insight into the general pricing trend. It's important to remember, however, that the mean can be affected by extremely high or low values鈥攌nown as outliers鈥攚hich might misrepresent the typical cost.
Median Calculation
The median value represents the middle point of a data set, meaning that half of the numbers are above the median and half below. To find the median, we arrange the tuna prices in ascending order and then locate the central number.

With an even number of values, like the 14 brands in our exercise, the median is the average of the two central numbers. Unlike the mean, the median is not influenced by outliers, making it a more accurate representation of what a 'typical' value might be for skewed data sets.
Data Skewness
Skewness in data indicates whether the distribution of values is symmetrical or if it tends to lean more heavily to one side. If the mean and median are different, this can hint at skewness. A larger mean compared to the median suggests a right or positive skew, whereas a smaller mean suggests a left or negative skew.

In examining skewness, it is crucial to look at the context and the differences between these two measures. The larger the gap, the more pronounced the skew. However, for subtle disparities, a graphical representation such as a histogram may offer better insight than numeric methods alone.
Probability and Statistics
The study of probability and statistics is key in making sense of data and the uncertainty inherent in it. Understanding mean and median is just the beginning鈥攊t enables us to summarize data concisely. Probability, on the other hand, helps us predict the likelihood of events based on a dataset.

Together, these concepts allow us to make informed decisions, whether we're choosing the most cost-effective brand of tuna or forecasting sales. Through probability and statistics, we learn not just to gather and describe data, but to draw meaningful conclusions and prepare for the future.

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Most popular questions from this chapter

You are given \(n=8\) measurements: \(3,1,5,6,\) 4,4,3,5 a. Calculate the range. b. Calculate the sample mean. c. Calculate the sample variance and standard deviation. d. Compare the range and the standard deviation. The range is approximately how many standard deviations?

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Construct a box plot for these data and identify any outliers: $$ 25,22,26,23,27,26,28,18,25,24,12 $$

The mean duration of television commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that durations are approximately normally distributed. a. What is the approximate probability that a commercial will last less than 35 seconds? b. What is the approximate probability that a commercial will last longer than 55 seconds?
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