Questions: Over the past several months, customers at Stymie's department store have been asked to fill out customer satisfaction surveys. In the surveys, the customers rate their shopping experience on a scale of 0 to 100. Here are 9 recent ratings. 86,55,86,75,40,55,72,86,55 (a) What is the mean of this data set? If your answer is not an integer, round your answer to one decimal place. (b) What is the median of this data set? If your answer is not an integer, round your answer to one decimal place. (c) How many modes does the data set have, and what are their values? Indicate the number of modes by clicking in the appropriate circle, and then indicate the value(s) of the mode(s), if applicable.

Over the past several months, customers at Stymie's department store have been asked to fill out customer satisfaction surveys. In the surveys, the customers rate their shopping experience on a scale of 0 to 100. Here are 9 recent ratings.
86,55,86,75,40,55,72,86,55
(a) What is the mean of this data set? If your answer is not an integer, round your answer to one decimal place.
(b) What is the median of this data set? If your answer is not an integer, round your answer to one decimal place.
(c) How many modes does the data set have, and what are their values? Indicate the number of modes by clicking in the appropriate circle, and then indicate the value(s) of the mode(s), if applicable.

Transcript text: Over the past several months, customers at Stymie's department store have been asked to fill out customer satisfaction surveys. In the surveys, the customers rate their shopping experience on a scale of 0 to 100 . Here are 9 recent ratings. \[ 86,55,86,75,40,55,72,86,55 \] (a) What is the mean of this data set? If your answer is not an integer, round your answer to one decimal place. (b) What is the median of this data set? If your answer is not an integer, round your answer to one decimal place. (c) How many modes does the data set have, and what are their values? Indicate the number of modes by clicking in the appropriate circle, and then indicate the value(s) of the mode(s), if applicable.

Solution

Solution Steps

To solve the given problem, we need to calculate the mean, median, and mode(s) of the provided data set.

Mean: Sum all the numbers and divide by the count of numbers.
Median: Sort the numbers and find the middle value. If the count of numbers is even, take the average of the two middle numbers.
Mode: Identify the number(s) that appear most frequently in the data set.

Step 1: Calculate the Mean

The mean of a data set is calculated by summing all the values and dividing by the number of values. Given the data set: \[ 86, 55, 86, 75, 40, 55, 72, 86, 55 \]

The sum of the values is: \[ 86 + 55 + 86 + 75 + 40 + 55 + 72 + 86 + 55 = 610 \]

The number of values is: \[ 9 \]

Thus, the mean is: \[ \text{Mean} = \frac{610}{9} \approx 67.7778 \]

Rounded to one decimal place: \[ \text{Mean} \approx 67.8 \]

Step 2: Calculate the Median

The median is the middle value of a data set when it is ordered from least to greatest. First, we sort the data set: \[ 40, 55, 55, 55, 72, 75, 86, 86, 86 \]

Since there are 9 values (an odd number), the median is the middle value: \[ \text{Median} = 72 \]

Step 3: Calculate the Mode(s)

The mode is the value(s) that appear most frequently in the data set. From the sorted data set: \[ 40, 55, 55, 55, 72, 75, 86, 86, 86 \]

We see that both 55 and 86 appear three times, which is more frequent than any other value. Therefore, there are two modes: \[ \text{Modes} = 55, 86 \]

The number of modes is: \[ \text{Number of modes} = 2 \]