The following data represent the number of live births in the United States in 2007 for women 15 to 49 years old.
Age Live Births
15 - 19 445,045
20 - 24 1,082,837
25 - 29 1,208,405
30 - 34 962,179
35 - 39 499,916
40 - 44 105,071
45 - 49 7,349
Find the following:
1. What is the class width?
2. What is the sample size (n)?
3. What is the mean of the ages?
4. What is the standard deviation of the ages?
5. Construct a relative frequency distribution.
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To find the answers to the questions, we will need to perform some calculations using the given data. Let's go through each question step by step:
1. To find the class width, we need to subtract the lower limit of one class from the lower limit of the next class. In this case, the classes are based on age groups. The class width will be the same for all age groups. So, let's calculate it:
- Class width = Lower limit of the next class - Lower limit of the current class
- For example, for the first class (15-19), the lower limit is 15 and the next class is 20-24. So, the class width will be 20 - 15 = 5.
- The class width for all age groups will be the same, which is 5.
2. To find the sample size (n), we need to sum up the live births for all age groups:
- n = Sum of live births for all age groups
- Add up all the values given for live births: 445,045 + 1,082,837 + 1,208,405 + 962,179 + 499,916 + 105,071 + 7,349 = 4,310,802.
- The sample size, n, is 4,310,802.
3. To find the mean of the ages, we need to calculate the weighted average of the age groups using the live births as weights:
- Mean = Sum of (Age * Live births) / n
- Multiply each age group by its respective live births and sum up the products:
[(15+19)/2 * 445,045] + [(20+24)/2 * 1,082,837] + [(25+29)/2 * 1,208,405] + [(30+34)/2 * 962,179] + [(35+39)/2 * 499,916] + [(40+44)/2 * 105,071] + [(45+49)/2 * 7,349]
- Divide the sum by the sample size (n) to get the mean.
4. To find the standard deviation of the ages, we need to calculate the sample standard deviation using the formula:
- Standard deviation = sqrt( (Sum of (Age - Mean)^2 * Live births) / (n - 1) )
- For each age group, subtract the mean from the age, square the difference, multiply by the respective live births, and sum up the products.
- Divide the sum by (n - 1) and take the square root to get the standard deviation.
5. To construct a relative frequency distribution, we need to calculate the proportion of live births in each age group compared to the total sample size (n):
- Relative frequency = Live births / n
- Divide each live births value by the sample size (n) to get the relative frequency.
By following these steps and performing the necessary calculations, you can find the answers to the given questions.