1. **State the problem:** We have a table showing the number of guests in different age groups at a wedding. We need to (a) draw a histogram, (b) find the median age group, (c) find the smallest possible number of guests under 50 years, and (d) estimate the mean age using mid-interval values. 2. **Draw the histogram:** - The x-axis represents age groups: 0-20, 20-40, 40-60, 60-80, 80-100. - The y-axis represents the number of guests, from 0 to 70 in steps of 10. - Bars heights correspond to the number of guests: 9, 58, 42, 27, 4 respectively. 3. **Find the median age group:** - Total guests = $9 + 58 + 42 + 27 + 4 = 140$. - Median position = $\frac{140 + 1}{2} = 70.5$th guest. - Cumulative counts: - Up to 0-20: 9 - Up to 20-40: $9 + 58 = 67$ - Up to 40-60: $67 + 42 = 109$ - The 70.5th guest lies in the 40-60 age group because 67 < 70.5 \leq 109. 4. **Smallest possible number of guests under 50 years:** - Guests under 50 include all in 0-20 and 20-40 groups plus some in 40-60 group. - Number in 0-20 and 20-40 = $9 + 58 = 67$. - The 40-60 group has 42 guests. - Since 50 is halfway in 40-60, the smallest number under 50 is if all guests from 40-60 are aged 50 or above except the minimum needed to reach 50. - To find how many guests are under 50, note that 50 is halfway in 40-60, so half of 42 = 21 guests could be under 50. - So smallest possible number under 50 = $67 + 0 = 67$ if none in 40-60 are under 50, but since median is in 40-60, at least some must be under 50. - More precisely, since median is 70.5th guest and cumulative up to 20-40 is 67, the 3.5 guests into 40-60 group are under 50. - So smallest possible number under 50 = $67 + 3.5 = 70.5$ guests. 5. **Estimate mean age using mid-interval values:** - Mid-intervals: 0-20: 10, 20-40: 30, 40-60: 50, 60-80: 70, 80-100: 90. - Multiply mid-interval by number of guests: - $10 \times 9 = 90$ - $30 \times 58 = 1740$ - $50 \times 42 = 2100$ - $70 \times 27 = 1890$ - $90 \times 4 = 360$ - Sum of products = $90 + 1740 + 2100 + 1890 + 360 = 6180$ - Mean age = $\frac{6180}{140} = 44.142857...$ - Rounded to 1 decimal place: 44.1 years. **Final answers:** - (b) Median age group: 40-60 years. - (c) Smallest possible number under 50 years: 70.5 guests. - (d) Estimated mean age: 44.1 years.