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Chapter 8: Problem 3

The numbers of incorrect answers on a true-false competency test for a random sample of 15 students were recorded as follows: 2,1,3,0,1,3,6,0,3,3,5 , \(2,1,4,\) and \(2 .\) Find (a) the mean; (b) the median; (c) the mode.

Short Answer

Expert verified
The mean (average) is 2.2, the median (middle score) is 2, and the mode (most common score) is 3.

Step by step solution

01

Calculate the Mean

The mean is the average of all the numbers in a data set. It can be calculated by adding all the numbers together, then dividing by the number of numbers. For this exercise, the mean can be calculated as follows: \( Mean = (2+1+3+0+1+3+6+0+3+3+5+2+1+4+2) / 15 \)
02

Determine the Median

The median is the middle value of a data set when the numbers are arranged in ascending order. If the data set has an odd number of observations, the median is the middle number. If the data set has an even number of observations, the median is the average of the two middle numbers. In this exercise, to find the median, arrange the data in ascending order, then determine the middle number. If there are two middle numbers, find their average.
03

Find the Mode

The mode is the number that appears most frequently in a data set. A set of data may have one mode, more than one mode, or no mode at all. Here, identify which number from the test scores appears most frequently.

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Key Concepts

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

Mean Calculation
Calculating the mean, often referred to as the average, is a fundamental concept in descriptive statistics. To find the mean, we sum up all the numbers in a dataset and then divide this total by the number of values. Let's consider our example with 15 students' test results, given as: 2, 1, 3, 0, 1, 3, 6, 0, 3, 3, 5, 2, 1, 4, and 2.
Here's a simple step-by-step approach:
  • First, add all the values together: \( 2 + 1 + 3 + 0 + 1 + 3 + 6 + 0 + 3 + 3 + 5 + 2 + 1 + 4 + 2 = 36 \)
  • Then, divide the total sum by the number of values, which in this case is 15: \( \text{Mean} = \frac{36}{15} \)
  • This results in a mean of 2.4.
The mean provides a single number that reflects the central tendency of the dataset, making it a useful measure for understanding overall performance.
Median Determination
The median is a measure of central tendency that indicates the middle point of a dataset. When calculating the median, the first step is to organize all numbers in ascending order. For our dataset of test scores, we rearrange it as follows: 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 5, 6.
To find the median:
  • Identify the middle number in the ordered list. With 15 numbers, the middle one is the 8th value.
  • In this ordered list, the 8th value is 2.
Thus, the median of this dataset is 2. The median gives us insights into the center of the data and is especially useful when the dataset includes outliers, as it is not influenced by extremely high or low values.
Mode Identification
The mode is the value that appears most frequently in a dataset. It can provide insights into what is most common or popular within the dataset. Unlike the mean and median, the mode can have multiple values or no mode at all if all numbers are unique.
For our student test results: 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 5, 6, finding the mode involves checking which number occurs most frequently:
  • Upon analyzing, we find the number 3 appears most frequently, occurring four times.
Hence, the mode of the dataset is 3. Understanding the mode can be valuable in identifying trends, helping to see the most frequent occurrence within the dataset.

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Most popular questions from this chapter

Define suitable populations from which the following samples are selected: (a) Persons in 200 homes are called by telephone in the city of Richmond and asked to name the candidate that they favor for election to the school board. (b) A coin is tossed 100 times and 34 tails are recorded. (c) Two hundred pairs of a new type of tennis shoe were tested on the professional tour and, on the average, lasted 4 months. (d) On five different occasions it took a lawyer \(21,26,\) \(24,22,\) and 21 minutes to drive from her suburban home to her midtown office.

Suppose a filling machine is used to fill cartons with a liquid product. The specification that is strictly enforced for the filling machine is \(9 \pm 1.5\) oz. If any carton is produced with weight outside these bounds, it is considered by the supplier to be a defective. It is hoped that at least \(99 \%\) of cartons will meet these specifications. With the conditions \(\mu=9\) and \(a=1\), what proportion of cartons from the process are defective? If changes are made to reduce variability, what must \(a\) be reduced to in order to meet specifications with probability \(0.99 ?\) Assume a normal distribution for the weight.

In a chemical process the amount of a certain type of impurity in the output is difficult to control and is thus a random variable. Speculation is that the population mean amount of the impurity is 0.20 grams per gram of output. It is known that the standard deviation is 0.1 grams per gram. An experiment is conducted to gain more insight regarding the speculation that \(\mu=0.2\). The process was run on a lab scale 50 times and the sample average \(x\) turned out to be 0.23 grams per gram. Comment on the speculation that the mean amount of impurity is 0.20 grams per gram. Make use of the central limit theorem in your work.

The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean \(\mu=3.2\) minutes and a standard deviation \(\sigma=1.6\) minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller's counter is (a) at most 2.7 minutes: (b) more than 3.5 minutes; (c) at least 3.2 minutes but less than 3.4 minutes.

A certain type of thread is manufactured with a mean tensile strength of 78.3 kilograms and a standard deviation of 5.6 kilograms. How is the variance of the sample mean changed when the sample size is (a) increased from 64 to \(196 ?\) (b) decreased from 784 to \(49 ?\)
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