This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Create an extended frequency distribution table. To find the mean and standard deviation, we need the class midpoints (x), the product of frequency and midpoint (fx), the square of the midpoint (x^2), and the product of frequency and squared midpoint (fx^2). For the median class, we need the cumulative frequency (cf). | Class Interval | Frequency (f) | Class Midpoint (x) | fx | Cumulative Frequency (cf) | x^2 | fx^2 | | :------------- | :-------------- | :------------------- | :--- | :-------------------------- | :---- | :----- | | 64-64 | 5 | 64 | 320 | 5 | 4096 | 20480 | | 65-69 | 3 | 67 | 201 | 8 | 4489 | 13467 | | 70-74 | 6 | 72 | 432 | 14 | 5184 | 31104 | | 75-79 | 4 | 77 | 308 | 18 | 5929 | 23716 | | 80-84 | 8 | 82 | 656 | 26 | 6724 | 53792 | | 85-89 | 10 | 87 | 870 | 36 | 7569 | 75690 | | 90-94 | 1 | 92 | 92 | 37 | 8464 | 8464 | | 95-99 | 3 | 97 | 291 | 40 | 9409 | 28227 | | Total | f = 40 | | fx = 3170 | | | fx^2 = 254940 | Step 2: Find the Modal Class. The modal class is the class interval with the highest frequency. From the table, the highest frequency is 10, which corresponds to the class interval 85-89. The modal class is 85-89. Step 3: Find the Median Class. The median class is the class where the cumulative frequency first exceeds or equals (N)/(2). The total number of data points is N = f = 40. We need to find the class containing the (N)/(2)-th value, which is the (40)/(2) = 20-th value. Looking at the cumulative frequency (cf) column: • The class 75-79 has a cf of 18. • The class 80-84 has a cf of 26. Since 26 is the first cumulative frequency greater than or equal to 20, the median class is 80-84. The median class is 80-84. Step 4: Calculate the Mean (x). The formula for the mean of grouped data is: x = ( fx)/(N) Substitute the values from the table: x = (3170)/(40) x = 79.25 The mean is 79.25. Step 5: Calculate the Standard Deviation (s). The formula for the standard deviation of grouped data is: s = sqrt(( fx^2 - ( fx)^2 / N)/(N-1)) Substitute the values from the table: fx^2 = 254940 fx = 3170 N = 40 s = sqrt((254940 - (3170)^2 / 40)/(40-1)) s = sqrt((254940 - 10048900 / 40)/(39)) s = sqrt((254940 - 251222.5)/(39)) s = sqrt((3717.5)/(39)) s = sqrt(95.32051282) s ≈ 9.76 The standard deviation is 9.76. That's 2 down. 3 left today — send the next one.