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Chapter 3: Problem 3

Data on manufacturing defects per 100 cars for the 30 brands of cars sold in the United States (USA TODAY, March 29,2016\()\) are: $$ \begin{array}{lllllllllll} 97 & 134 & 198 & 142 & 95 & 135 & 132 & 145 & 136 & 129 & 152 \\ 158 & 169 & 155 & 106 & 125 & 120 & 153 & 208 & 163 & 204 & 173 \\ 165 & 126 & 113 & 167 & 171 & 166 & 181 & 161 & & & \end{array} $$

Short Answer

Expert verified
In conclusion, the mean is 153, the median is 154, there is no mode, the range is 113, the variance is approximately 4767.31, and the standard deviation is approximately 69.04.

Step by step solution

01

Arrange the data in ascending order

First, we arrange the given data in ascending order: $$ \begin{array}{lllllllllll} 95 & 97 & 106 & 113 & 120 & 125 & 126 & 129 & 132 & 134 & 135 \\\ 136 & 142 & 145 & 152 & 153 & 155 & 158 & 161 & 163 & 165 & 166 \\\ 167 & 169 & 171 & 173 & 181 & 198 & 204 & 208 & & & \end{array} $$
02

Calculate mean

To find the mean, we sum up all the values and divide by the total number of values. There are 30 values in this dataset, so: Mean = \( \frac{95+97+106+\cdots+204+208}{30} \) Mean = \( \frac{4589}{30} \) Mean = 153
03

Calculate median

As the data is now in ascending order, we can easily find the median by locating the middle value. In this case, since there are 30 values, we take the average of the 15th and 16th values, which are 153 and 155: Median = \( \frac{153+155}{2} = 154 \)
04

Calculate mode

The mode is the value that appears most frequently in the dataset. By looking at the dataset, we don't see any repeating values: Mode = None
05

Calculate range

The range is the difference between the largest and smallest values: Range = 208 - 95 = 113
06

Calculate variance

To calculate variance, we first need to calculate the sum of the squared deviations from the mean: Sum of squared deviations = \( (95-153)^2 + (97-153)^2 + \cdots + (204-153)^2 + (208-153)^2 \) Sum of squared deviations = 138262 Now, we divide this sum by the number of values minus one, i.e., 30 - 1 = 29: Variance = \( \frac{138262}{29} \approx 4767.31 \)
07

Calculate standard deviation

The standard deviation is the square root of the variance: Standard deviation = \( \sqrt{4767.31} \approx 69.04 \) In conclusion, the mean is 153, the median is 154, there is no mode, the range is 113, the variance is approximately 4767.31, and the standard deviation is approximately 69.04.

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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 referred to as the average, is a fundamental concept in descriptive statistics. It represents the typical value in a dataset by summing up all the numbers and dividing by the count of values. For example, when assessing manufacturing defects per 100 cars, the mean tells us the average number of defects.

To calculate the mean, you follow these steps:
  • Add up all the values in the dataset.
  • Divide the sum by the total number of values.
In our case, we sum up the number of defects for the 30 brands and divide by 30, giving us a mean of 153 defects. This is a crucial piece of information for manufacturers aiming to understand overall quality control.
Median Calculation
The median is the middle value in a dataset when arranged in ascending or descending order. Unlike the mean, the median is not affected by extremely high or low values, which makes it a valuable measure of central tendency, especially in skewed distributions.

Here's how to find the median:
  • Arrange the data from lowest to highest.
  • If the number of observations is odd, the median is the middle number.
  • If the number of observations is even, as in our car defect example, average the two central numbers.
For the defects per 100 cars, we arranged 30 data points in order and found the middle by averaging the 15th and 16th values, yielding a median of 154 defects. This represents the center of our data set.
Variance and Standard Deviation
Variance and standard deviation are both measures of spread in a dataset, describing how far the data points are from the mean.

Variance is the average of the squared differences from the Mean. To calculate it, we:
  • Determine the mean.
  • Subtract the mean from each data point and square the result.
  • Sum all the squared results.
  • Divide by the number of data points minus one (this is the Sample Variance).
Standard deviation is the square root of the variance, providing a measure of spread in the same units as the data itself. For the car defects dataset, with a variance of approximately 4767.31, the standard deviation comes out to approximately 69.04. This tells us that, on average, the number of defects per 100 cars deviates from the mean (153) by about 69 cars.

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Most popular questions from this chapter

Data on weekday exercise time for 20 males, consistent with summary quantities given in the paper "An Ecological Momentary Assessment of the Physical Activity and Sedentary Behaviour Patterns of University Students" (Health Education Journal [2010]\(: 116-125),\) are shown below. Calculate and interpret the values of the median and interquartile range. $$ \begin{array}{rrrrrrrrr} 43.5 & 91.5 & 7.5 & 0.0 & 0.0 & 28.5 & 199.5 & 57.0 & 142.5 \\ 8.0 & 9.0 & 36.0 & 0.0 & 78.0 & 34.5 & 0.0 & 57.0 & 151.5 \\ 8.0 & 0.0 & & & & & & & \end{array} $$

Morningstar is an investment research firm that publishes some online educational materials. The materials for an online course called "Looking at Historical Risk" (news.morningstar.com/classroom2/course .asp?docld \(=2927 \&\) page \(=2 \& \mathrm{CN}=\mathrm{com},\) retrieved August 3,2016\()\) included the following paragraph referring to annual return (in percent) for investment funds: Using standard deviation as a measure of risk can have its drawbacks. It's possible to own a fund with a low standard deviation and still lose money. In reality, that's rare. Funds with modest standard deviations tend to lose less money over short time frames than those with high standard deviations. For example, the one-year average standard deviation among ultrashort-term bond funds, which are among the lowest-risk funds around (other than money market funds), is a mere \(0.64 \%\). a. Explain why the standard deviation of percent return is a reasonable measure of unpredictability and why a smaller standard deviation for a funds percent return means less risk. b. Explain how a fund with a small standard deviation can still lose money. (Hint: Think about the average percent return.)

The paper referenced in the previous exercise also gave data on the actual amount (in \(\mathrm{ml}\) ) poured into a short, wide glass for individuals asked to pour 1.5 ounces \((44.3 \mathrm{ml})\) $$ \begin{array}{llllllll} 89.2 & 68.6 & 32.7 & 37.4 & 39.6 & 46.8 & 66.1 & 79.2 \\ 66.3 & 52.1 & 47.3 & 64.4 & 53.7 & 63.2 & 46.4 & 63.0 \\ 92.4 & 57.8 & & & & & & \end{array} $$

The mean playing time for a large collection of compact discs is 35 minutes, and the standard deviation is 5 minutes. a. What value is 1 standard deviation above the mean? One standard deviation below the mean? What values are 2 standard deviations away from the mean? b. Assuming that the distribution of times is mound shaped and approximately symmetric, approximately what percentage of times are between 25 and 45 minutes? Less than 20 minutes or greater than 50 minutes? Less than 20 minutes? (Hint: See Example \(3.19 .\) )

The accompanying data on total amount of time per day (in minutes) spent using a cell phone are consistent with summary statistics in the paper "The Relationship Between Cell Phone Use and Academic Performance in a Sample of U.S. College Students" (SAGE Open [2015]: 1-9). $$ \begin{array}{rrrrrrrrr} 225 & 318 & 468 & 0 & 236 & 601 & 144 & 196 & 374 \\ 0 & 424 & 198 & 156 & 734 & 331 & 502 & 0 & 492 \\ 563 & 195 & 237 & 110 & 516 & 422 & 740 & & \end{array} $$ Calculate and interpret the values of the median and the interquartile range.
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