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Data on a customer satisfaction rating (called the APEAL rating) are given for each brand of car sold in the United States (USA Today, July 17,2010 ). The APEAL rating is a score between 0 and 1,000 , with higher values indicating greater satisfaction. \(\begin{array}{lllllllll}822 & 832 & 845 & 802 & 818 & 789 & 748 & 751 & 794 \\ 792 & 766 & 760 & 805 & 854 & 727 & 761 & 836 & 822 \\\ 820 & 774 & 842 & 769 & 815 & 767 & 763 & 877 & 780 \\ 764 & 755 & 750 & 745 & 797 & 795 & & & \end{array}\) Calculate and interpret the mean and standard deviation for this data set.

Short Answer

Expert verified
The mean and the standard deviation are numeric summaries that give an idea of the general satisfaction level (mean), and how much the scores differ from each other and from the mean (standard deviation). Specific values will be determined by the calculation.

Step by step solution

01

Calculate the mean

Firstly, add up all the APEAL ratings provided in the data set and then divide by the total number of ratings to get the mean (average). This can be done using the formula for the mean, which is given by: \[ mean = \frac{\sum x_i}{n} \], where \(x_i\) represents each individual score and \(n\) is the total number of scores.
02

Calculate the standard deviation

To calculate the standard deviation, first calculate the variance, which is the average of the squared differences from the mean. This is done by subtracting the mean from each score, squaring the result, and then averaging these squared differences. This can be calculated using the formula for variance: \[ variance = \frac{\sum (x_i - mean)^2}{n} \]. After the variance is calculated, take the square root to find the standard deviation. The formula for the standard deviation is the square root of variance: \[ standard_deviation = \sqrt{variance} \]. The standard deviation gives a measure of how much the values in the dataset deviate from the mean.
03

Interpret the Results

The mean score provides an overall measure of customer satisfaction, while the standard deviation indicates the level of variation in the scores. A high standard deviation would suggest that the satisfaction levels vary greatly among the customers, whereas a low standard deviation would indicate that the satisfaction scores are closely packed around the average value.

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Key Concepts

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

Mean Calculation
The mean, or average, is one of the simplest yet most essential concepts in statistics. It provides a single value representing the center point of a data set. To calculate the mean, you sum up all the individual data points and divide this total by the number of points. In the context of customer satisfaction ratings, the mean gives us an insight into the overall satisfaction level across all brands of cars.

For our exercise, the mean satisfaction score is found using the formula \[ mean = \frac{\sum x_i}{n} \], where \(x_i\) are the satisfaction scores, and \(n\) is the total number of scores. By calculating the mean, businesses can assess their performance in terms of customer satisfaction and benchmark this against competitors or industry averages.
Standard Deviation
Standard deviation is a powerful statistical tool that measures the amount of variability or dispersion in a set of values. It tells us how spread out the data points are in relation to the mean. A small standard deviation indicates that the data points tend to be close to the mean, while a large standard deviation indicates that the data points are widespread.

To find the standard deviation, we first need to calculate the variance. Then, we take the square root of that variance. The formula \[ standard\_deviation = \sqrt{variance} \] aids in understanding the consistency of customer satisfaction. For example, a low standard deviation would suggest that most customers rate their satisfaction similarly, thus pointing to uniformity in the perceived quality of the cars.
Variance
Variance is a statistic that represents the degree to which each number in a dataset differs from the mean and hence from every other number in the dataset. It is essentially the average of the squared differences between each data point and the mean. To compute it, you subtract the mean from each score, square the result, and then average these squared differences using the formula \[ variance = \frac{\sum (x_i - mean)^2}{n} \].

In the real world, a high variance within customer satisfaction ratings might indicate inconsistent experiences or a polarized customer base, where some are very satisfied and others are not. Companies typically aim for a lower variance, striving for consistent and predictable customer satisfaction.
Data Interpretation
After crunching numbers, data interpretation comes into play, which is the process of making sense of numerical findings to make informed decisions. For the customer satisfaction ratings, interpreting the mean tells us about the general level of satisfaction while the standard deviation offers insights into the variability of customers' perceptions.

These metrics become particularly valuable when compared across different time periods or different demographics. For instance, if the standard deviation of satisfaction scores increases over time, it might suggest that customer experiences are becoming more diverse, possibly due to changes in products or services. Ultimately, this statistical analysis is crucial as it supports decision-making that can enhance customer satisfaction and business performance overall.

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Most popular questions from this chapter

The San Luis ObispoTelegram-Tribune(October1,1994) reported the following monthly salaries for supervisors from six different counties: \(\$ 5,354\) (Kern), \(\$ 5,166\) (Monterey), \(\$ 4,443\) (Santa Cruz), \(\$ 4,129\) (Santa Barbara), \(\$ 2,500\) (Placer), and \$2,220 (Merced). San Luis Obispo County supervisors are supposed to be paid the average of the two counties in the middle of this salary range. Which measure of center determines this salary, and what is its value? Find the value of the other measure of center featured in this chapter. Why is it not as favorable to the San Luis Obispo County supervisors (although it might appeal to taxpayers)?

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