{"id":"item-4","content":"Mark the class intervals on the horizontal axis and frequency density on the vertical axis."},{"id":"item-5","content":"Draw touching bars with the correct widths and heights."}],"order":8,"instruction":"Put these steps for completing and drawing a histogram in the correct order.","correctOrder":["item-1","item-2","item-3","item-4","item-5"]},{"id":"card-9","type":"content","order":9,"title":"Worked example: complete the missing values","blocks":[{"id":"block-1","content":"A table gives $0 \\le t < 4$ frequency 10, $4 \\le t < 9$ frequency density 1.4, $9 \\le t < 15$ frequency density 1.5, and $15 \\le t < 25$ frequency 12. First find the class widths: $4$, $5$, $6$, and $10$."},{"id":"block-2","content":"For the classes with known frequencies, divide by the width to get the bar heights:\n$$\n\\begin{aligned}\n\\text{f.d. for }0 \\le t < 4 &= \\frac{10}{4}=2.5\\\\\n\\text{f.d. for }15 \\le t < 25 &= \\frac{12}{10}=1.2\n\\end{aligned}\n$$\nThese are the heights of those two bars."},{"id":"block-3","content":"For the classes with known frequency densities, multiply by the width to get the frequencies:\n$$\n\\begin{aligned}\n\\text{frequency for }4 \\le t < 9 &= 5 \\times 1.4=7\\\\\n\\text{frequency for }9 \\le t < 15 &= 6 \\times 1.5=9\n\\end{aligned}\n$$\nThese are the missing frequencies."},{"id":"block-4","content":"So the completed rows are frequency 10 and frequency density 2.5, frequency 7 and frequency density 1.4, frequency 9 and frequency density 1.5, and frequency 12 and frequency density 1.2. Draw touching bars across the correct intervals, with heights $2.5$, $1.4$, $1.5$, and $1.2$ on the frequency density axis."}]},{"id":"card-10","type":"fill_blank","order":10,"blanks":[{"id":"w","isMath":true,"correctAnswer":"10"},{"id":"fd","isMath":true,"correctAnswer":"1.2"}],"sentence":"For $15 \\le t < 25$ with frequency 12, the class width is {{blank:w}} and the frequency density is {{blank:fd}}."},{"id":"card-11","type":"content","order":11,"title":"Estimating from part of a bar","blocks":[{"id":"block-1","content":"![Histogram bar from 10 to 20 with a highlighted sub-interval from 12 to 18 and height 2.5](https://assets.yesgenie.com/lesson-v2-img/inline/d1827ecd-04a1-421f-b748-da5807ca8499.jpeg)\nSometimes you only want part of a class interval, such as $12 \\le t < 18$ inside the bar $10 \\le t < 20$. You use the same frequency density within that bar and assume the data is spread evenly through the class."},{"id":"block-2","content":"The smaller interval has width $18 - 12 = 6$. So the estimated frequency is\n$$\n6 \\times 2.5 = 15\n$$\nThe estimate is 15."},{"id":"block-3","content":"This is only an estimate because the real data might not be perfectly even inside the class. The method works by taking the area of just the highlighted part of the bar."}]},{"id":"card-12","type":"mcq","order":12,"options":[{"id":"a","text":"5","isCorrect":false},{"id":"b","text":"8","isCorrect":true},{"id":"c","text":"16","isCorrect":false},{"id":"d","text":"24","isCorrect":false}],"question":"A class interval $20 \\le x < 35$ has frequency 24. Estimate the frequency for $26 \\le x < 31$.","explanation":"The full class width is $35 - 20 = 15$, so the frequency density is $24 \\div 15 = 1.6$. The smaller interval has width 5, so the estimate is $5 \\times 1.6 = 8$; the common wrong answer C comes from using the wrong fraction of the class.","correctOptionId":"b"},{"id":"card-13","type":"content","order":13,"title":"Common mistakes and exam checks","blocks":[{"id":"block-1","content":"Read the vertical axis carefully. If it says frequency density, a bar with width 10 and height 2 has frequency $10 \\times 2 = 20$."},{"id":"block-2","content":"Write each class width beside the interval before you calculate. This stops two common errors: drawing all bars the same width and using raw frequency as the bar height."},{"id":"block-3","content":"Bars should touch because the data is continuous. If you use only part of a bar, say that the answer is an estimate."}]},{"id":"card-14","type":"multi-question","order":14,