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Chapter 3: Problem 21

Nixon Corporation manufactures computer monitors. The following data are the numbers of computer monitors produced at the company for a sample of 10 days. \(\begin{array}{lllllll}24 & 32 & 27 & 23 & 35 & 33 & 29\end{array}\) 40 23 28 Calculate the mean, median, and mode for these data.

Short Answer

Expert verified
The mean number of monitors produced over the 10 day period is 30.4, the median is 28.5, and the mode (most commonly produced number of monitors in a day) is 23.

Step by step solution

01

Calculating the Mean

Add all the values in the data sample together and then divide by the number of values. That is, \( Mean = \frac{(24+32+27+23+35+33+29+40+23+28)}{10} = 30.4 \).
02

Calculating the Median

To find the median, first arrange the data in ascending order: \(\begin{array}{lllllll}23, 23, 24, 27, 28, 29, 32, 33, 35, 40\end{array}\). The number of data points is even (10), so the median will be the mean of the two middle numbers (values at the 5th and 6th place). That is \( Median = \frac{(28+29)}{2} = 28.5 \).
03

Calculating the Mode

The mode is the number occurring most frequently in a data set. Looking at the ordered data, it can be seen that number '23' appears twice, and the rest of the numbers appear only once. Thus, the mode is 23.

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Key Concepts

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

Mean Calculation
The mean, often called the average, is a fundamental concept in statistics. It helps to summarize a set of numbers with a single value, offering a comprehensive picture of the data's central tendency. To find the mean, follow these simple steps:
  • Add up all the numbers in your data set. In our case, the numbers are: 24, 32, 27, 23, 35, 33, 29, 40, 23, and 28. The total sum is 304.
  • Count how many numbers (data points) you have. Here, there are 10 numbers.
  • Divide the total sum by the number of data points: \[Mean = \frac{304}{10} = 30.4\]
Hence, the mean production of computer monitors is 30.4 units per day.Remember, the mean gives a good overall picture, but it can be affected by very high or low values, often called outliers.
Median Calculation
The median is another measure of central tendency, often preferable when dealing with skewed data or outliers. It represents the middle value in a data set when the numbers are sorted in ascending order. Here's how you can determine it:
  • First, sort your data from smallest to largest: 23, 23, 24, 27, 28, 29, 32, 33, 35, 40.
  • If you have an odd number of data points, the median is the middle one. But, with an even number like ours (10), the median is the average of the two central numbers.
  • Locate the middle numbers. Here, the 5th and 6th numbers are 28 and 29.
  • Calculate their average: \[Median = \frac{28 + 29}{2} = 28.5\]
So, the median number of monitors produced daily is 28.5.The median is less sensitive to extremes than the mean, making it a reliable middle-point indicator for the data.
Mode Calculation
The mode is a simple yet powerful statistical measure indicating the most frequently occurring value in a data set. It is particularly useful for understanding which values are dominant within a set. Here鈥檚 how to find it:
  • Organize your data and look for any repeating numbers: 23, 23, 24, 27, 28, 29, 32, 33, 35, 40.
  • Identify the number that appears most often. In this example, the number 23 appears twice, while all other numbers appear only once.
Thus, the mode of this set is 23, implying that 23 is the most frequently produced number of monitors on any given day. In some data sets, there may be more than one mode, or none at all, if all numbers occur with the same frequency.

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Most popular questions from this chapter

One property of the mean is that if we know the means and sample sizes of two (or more) data sets, we can calculate the combined mean of both (or all) data sets. The combined mean for two data sets is calculated by using the formula $$ \text { Combined mean }=\bar{x}=\frac{n_{1} \bar{x}_{1}+n_{2} \bar{x}_{2}}{n_{1}+n_{2}} $$ where \(n_{1}\) and \(n_{2}\) are the sample sizes of the two data sets and \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the means of the two data sets, respectively. Suppose a sample of 10 statistics books gave a mean price of \(\$ 140\) and a sample of 8 mathematics books gave a mean price of \(\$ 160\). Find the combined mean. (Hint: For this example: \(\left.n_{1}=10, n_{2}=8, \bar{x}_{1}=\$ 140, \bar{x}_{2}=\$ 160 .\right)\)

The following data give the weights (in pounds) lost by 15 members of a health club at the end of 2 months after joining the club. \(\begin{array}{rrrrrrrr}5 & 10 & 8 & 7 & 25 & 12 & 5 & 14 \\ 11 & 10 & 21 & 9 & 8 & 11 & 18 & \end{array}\) a. Compute the values of the three quartiles and the interquartile range. b. Calculate the (approximate) value of the 82 nd percentile. c. Find the percentile rank of 10 .

On a 300 -mile auto trip, Lisa averaged 52 mph for the first 100 miles, 65 mph for the second 100 miles, and 58 mph for the last 100 miles. a. How long did the 300 -mile trip take? b. Could you find Lisa's average speed for the 300 -mile trip by calculating \((52+65+58) / 3 ?\) If not, find the correct average speed for the trip.

According to Fair Isaac, "The Median FICO (Credit) Score in the U.S. is 723" (The Credit Scoring Site, 2009). Suppose the following data represent the credit scores of 22 randomly selected loan applicants. \(\begin{array}{lllllllllll}494 & 728 & 468 & 533 & 747 & 639 & 430 & 690 & 604 & 422 & 356 \\ 805 & 749 & 600 & 797 & 702 & 628 & 625 & 617 & 647 & 772 & 572\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value 617 fall in relation to these quartiles? b. Find the approximate value of the 30 th percentile. Give a brief interpretation of this percentile. c. Calculate the percentile rank of 533 . Give a brief interpretation of this percentile rank.

The prices of all college textbooks follow a bell-shaped distribution with a mean of \(\$ 105\) and a standard deviation of \(\$ 20\). a. Using the empirical rule, find the percentage of all college textbooks with their prices between i. \(\$ 85\) and \(\$ 125\) ii. \(\$ 65\) and \(\$ 145\) \({ }^{*} \mathrm{~b}\). Using the empirical rule, find the interval that contains the prices of \(99.7 \%\) of college textbooks.
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