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Chapter 8: Problem 49

Use the following information to answer the next 13 exercises: The data in Table 8.10 are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag. $$\begin{array}{|c|c|}\hline X & {\text { Freq }} \\ \hline 1 & {1} \\\ \hline 2 & {7} \\ \hline 3 & {78} \\ \hline 4 & {7} \\ \hline 5 & {6} \\\ \hline\end{array}$$ Calculate the following: a. \(\overline{x}=\) b. \(s_{x}=\) c. \(n=\)

Short Answer

Expert verified
a. \(\overline{x} \approx 3.101\) b. \(s_x \approx 2.641\) c. \(n = 99\)

Step by step solution

01

Identify the Data for Calculation

We are given a frequency distribution for the number of colors on a flag. The frequencies are:\[ f_1 = 1, f_2 = 7, f_3 = 78, f_4 = 7, f_5 = 6 \] corresponding to the number of colors (\(X\)) which are 1, 2, 3, 4, and 5 respectively.
02

Calculate the Total Number of Flags

The total number of flags \(n\) is the sum of all frequencies:\[ n = f_1 + f_2 + f_3 + f_4 + f_5 = 1 + 7 + 78 + 7 + 6 = 99 \]
03

Calculate the Mean (\(\overline{x}\))

The mean \(\overline{x}\) is calculated using the formula:\[ \overline{x} = \frac{\sum (X_i \times f_i)}{n} \]Calculate each term:- For 1 color: \(1 \times 1 = 1\)- For 2 colors: \(2 \times 7 = 14\)- For 3 colors: \(3 \times 78 = 234\)- For 4 colors: \(4 \times 7 = 28\)- For 5 colors: \(5 \times 6 = 30\)Sum these products: \(1 + 14 + 234 + 28 + 30 = 307\)Then, apply the formula:\[ \overline{x} = \frac{307}{99} \approx 3.101 \]
04

Calculate the Standard Deviation (\(s_x\))

The standard deviation \(s_x\) is given by:\[ s_x = \sqrt{\frac{\sum (X_i^2 \times f_i) - n \cdot \overline{x}^2}{n - 1}} \]Calculate each term involving squares:- For 1 color: \(1^2 \times 1 = 1\)- For 2 colors: \(2^2 \times 7 = 28\)- For 3 colors: \(3^2 \times 78 = 702\)- For 4 colors: \(4^2 \times 7 = 112\)- For 5 colors: \(5^2 \times 6 = 150\)Sum these squared products: \(1 + 28 + 702 + 112 + 150 = 993\)Now substitute into the standard deviation formula:\[ s_x = \sqrt{\frac{993 - 99 \cdot 3.101^2}{98}} \approx \sqrt{\frac{993 - 309.2201}{98}} \approx \sqrt{\frac{683.7799}{98}} \approx \sqrt{6.976}\approx 2.641 \]

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Key Concepts

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

Frequency Distribution
In statistics, a frequency distribution is a powerful tool. It allows us to see how often values occur within a data set. In our exercise, the frequency distribution shows how many national flags have a certain number of colors.

For example, one flag has 1 color, seven flags have 2 colors, and so on. This shows the frequency (or how common) each number of colors is, which is crucial for analysis.

Understanding this distribution is the backbone for calculating things like the mean and standard deviation. It helps organize data into a clear view. Without it, we'd struggle to find patterns or make predictions.
Mean Calculation
The mean, also known as the average, is a measure of central tendency. It tells us what the central or typical value is in a data set.

In the exercise, to find the mean, you multiply each number of colors (8 extsubscript{i}) by their corresponding frequencies (8 extsubscript{i}) to get a total for each, then sum these totals. You get 307 in this case.

Divide this sum by the total number of flags (99) to find the mean: 4.18. This gives insight into the most "typical" number of colors on a flag. Calculating the mean allows for easy comparison between different data sets.
Standard Deviation
Standard deviation tells us about the amount of variation or spread in a dataset. It answers the question, "How much do values deviate from the mean?"

In the given exercise, the standard deviation is found by first calculating the squared deviations of each number of colors from the mean, multiplying by their frequencies, and summing them up. This provides a sense of how spread out the colors are across the flags.

Our calculation results in a standard deviation of approximately 2.641. This means the number of colors on a flag deviates from the mean by about 2.641 colors on average. Understanding this spread is important for statistical analysis and making confident predictions.
Random Survey
A random survey is used to gather data in a way that avoids bias. This method ensures that every individual in a population has an equal chance of being selected. When you perform a random survey, the collected data can more accurately represent the whole population.

In the context of our exercise, a random survey of national flags was conducted. This means the 39 flags were picked in a manner such that each flag had an equal chance of being chosen.

This method enhances the credibility of the statistical findings, allowing conclusions drawn from the frequency distribution and analysis to be more reliable. The randomness ensures that our results are not skewed and that they provide a valid insight into how many colors are typically found on flags globally.

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