Concept explainers
Many times in statistics it is necessary to see if a set of data values is approximately
The numbers of branches of the 50 top banks are shown.
1. Construct a frequency distribution for the data.
2. Construct a histogram for the data.
3. Describe the shape of the histogram.
4. Based on your answer to question 3, do you feel that the distribution is approximately normal?
In addition to the histogram, distributions that are approximately normal have about 68% of the values fall within 1 standard deviation of the
5. Find the mean and standard deviation for the data.
6. What percent of the data values fall within 1 standard deviation of the mean?
7. What percent of the data values fall within 2 standard deviations of the mean?
8. What percent of the data values fall within 3 standard deviations of the mean?
9. How do your answers to questions 6, 7, and 8 compare to 68, 95, and 100%, respectively?
10. Does your answer help support the conclusion you reached in question 4? Explain.
1.
![Check Mark](/static/check-mark.png)
To construct: The frequency distribution for the data.
Answer to Problem 1AC
The frequency distribution for the data is,
Limits | Frequency |
0-9 | 1 |
10-19 | 14 |
20-29 | 17 |
30-39 | 7 |
40-49 | 3 |
50-59 | 2 |
60-69 | 2 |
70-79 | 1 |
80-89 | 2 |
90-99 | 1 |
Explanation of Solution
Given info:
The data shows that the number of branches of the 50 top banks.
Calculation:
Answers may vary; one of the possible answers is given below.
First consider the limits with the difference of 9.
The limits are 0-9, 10-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89 and 90-99.
From the data, it can be observed that the number banks having branches between 0-9 is 1. Also, the number banks having branches between 10-19 is 14.
Similarly, the frequencies of other classes are obtained as follows.
Limits | Frequency |
0-9 | 1 |
10-19 | 14 |
20-29 | 17 |
30-39 | 7 |
40-49 | 3 |
50-59 | 2 |
60-69 | 2 |
70-79 | 1 |
80-89 | 2 |
90-99 | 1 |
2.
![Check Mark](/static/check-mark.png)
To construct: The histogram for the data.
Answer to Problem 1AC
The histogram for the data is,
Explanation of Solution
Calculation:
Step-by-step procedure to obtain histogram using the MINITAB software:
- Choose Graph > Histogram.
- Choose Simple, and then click OK.
- In Graph variables, enter the column of Number of branches of the bank.
- Click OK.
3.
![Check Mark](/static/check-mark.png)
To describe: The shape of the histogram.
Answer to Problem 1AC
The shape of the histogram is unimodel and skewed to the right.
Explanation of Solution
Justification:
From the histogram of part (2), it can be observed that the data has single group and the bar in histogram starts with a high value which is gradually decreasing to the minimum value. Thus, it can be conclude that shape of the histogram is unimodel and skewed to the right.
4.
![Check Mark](/static/check-mark.png)
To check: Whether the distribution is approximately normal or not based on the answer of question 3.
Answer to Problem 1AC
No, the distribution is not approximately normal.
Explanation of Solution
Justification:
From the part (3), it can be observed shape of the histogram is unimodel and skewed to the right. Thus, it can be conclude that the distribution does not appear to be normal.
5.
![Check Mark](/static/check-mark.png)
To find: The mean and standard deviation for the data.
Answer to Problem 1AC
The mean and standard deviation for the data are 31.4 and 20.6.
Explanation of Solution
Calculation:
- Choose Stat > Basic Statistics > Display Descriptive Statistics.
- In Variables enter the columns Variable.
- Choose option statistics, and select Mean and Standard deviation.
- Click OK.
Output using Minitab software is,
Form the Minitab output, it can be observed that the mean and standard deviation for the data are approximately 31.4 and 20.6.
6.
![Check Mark](/static/check-mark.png)
To find: The percent of the data values fall within 1 standard deviation of the mean.
Answer to Problem 1AC
The percent of the data values fall within 1 standard deviation of the mean is 80%.
Explanation of Solution
Calculation:
Here, 1 standard deviation of the mean represents that
Substitute 31.4 for
For
For
The percentage of the data values fall within 1 standard deviation of the mean represents the percentage of the data values lies between 10.8 and 52.
The formula for the percentage of the data values fall within 1 standard deviation of the mean is,
Substitute 40 for “Number of data values lies within 10.8 and 52” and 50 for “Total number of data values”
Thus, the percent of the data values fall within 1 standard deviation of the mean is 80%.
7.
![Check Mark](/static/check-mark.png)
To find: The percent of the data values fall within 2 standard deviation of the mean.
Answer to Problem 1AC
The percent of the data values fall within 2 standard deviation of the mean is 92%.
Explanation of Solution
Calculation:
Here, 2 standard deviation of the mean represents that
Substitute 31.4 for
For
For
The percentage of the data values fall within 2 standard deviation of the mean represents the percentage of the data values lies between –9.8 and 72.6.
The formula for the percentage of the data values fall within 2 standard deviation of the mean is,
Substitute 46 for “Number of data values lies within –9.8 and 72.6” and 50 for “Total number of data values”
Thus, the percent of the data values fall within 2 standard deviation of the mean is 92%.
8.
![Check Mark](/static/check-mark.png)
To find: The percent of the data values fall within 3 standard deviation of the mean.
Answer to Problem 1AC
The percent of the data values fall within 3 standard deviation of the mean is 98%.
Explanation of Solution
Calculation:
Here, 3 standard deviation of the mean represents that
Substitute 31.4 for
For
For
The percentage of the data values fall within 3 standard deviation of the mean represents the percentage of the data values lies between –30.4 and 93.2.
The formula for the percentage of the data values fall within 3 standard deviation of the mean is,
Substitute 49 for “Number of data values lies within –30.4 and 93.2” and 50 for “Total number of data values”
Thus, the percent of the data values fall within 3 standard deviation of the mean is 98%.
9.
![Check Mark](/static/check-mark.png)
To compare: The 6,7 and 8 answers with the 68, 95 and 100%.
Answer to Problem 1AC
The 6,7 and 8 answers are different with the 68, 95 and 100%.
Explanation of Solution
Calculation:
From the answers of 6,7 and 8, it is observed that the percent of the data values fall within 1,2 and 3 standard deviation of the mean are 80%, 92% and 98%. Thus, it can be conclude that the 6,7 and 8 answers are different with the 68, 95 and 100%.
10.
![Check Mark](/static/check-mark.png)
To explain: Whether the answer help to support the conclusion in question 4.
Answer to Problem 1AC
Yes, the answer help to support the conclusion in question 4 because the distribution of the variable does not follow normal distribution.
Explanation of Solution
Calculation:
From the answer of (9), it can be observed that the distribution of the variable does not follow normal distribution. Thus, it can be concluding that the answer help to support the conclusion in question 4.
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