1. **State the problem:** Given remission times for 12 patients in the placebo group: 22, 8, 12, 3, 17, 5, 11, 2, 15, 4, 8, 1, we need to perform statistical analyses including computing the mean, variance, standard deviation, z scores, percentiles, and interquartile range. 2. **Calculate the mean** $\bar{x}$: $$\bar{x} = \frac{22+8+12+3+17+5+11+2+15+4+8+1}{12} = \frac{108}{12} = 9$$ 3. **Complete the table for each $x$: calculate $x-\bar{x}$ and $(x - \bar{x})^2$:** - For example, for $x=22$, $22 - 9 = 13$, and $13^2 = 169$. Similarly for other values as given. - Total sum of deviations $\sum (x-\bar{x})=0$ (which must always be zero). - Total sum of squared deviations $\sum (x-\bar{x})^2 = 474$. 4. **Calculate variance $s^2$:** Variance formula for sample: $$s^2 = \frac{\sum (x - \bar{x})^2}{n-1} = \frac{474}{12-1} = \frac{474}{11} \approx 43.09$$ 5. **Calculate standard deviation $s$:** $$s = \sqrt{s^2} = \sqrt{43.09} \approx 6.56$$ 6. **Calculate Coefficient of Variation (CoV):** $$\text{CoV} = \frac{s}{\bar{x}} \times 100 = \frac{6.56}{9} \times 100 \approx 72.89$$ 7. **Calculate the z-score for highest value $x=22$:** $$z = \frac{x - \bar{x}}{s} = \frac{22 - 9}{6.56} = \frac{13}{6.56} \approx 1.98$$ - Since $z = 1.98$ lies within $-2$ and $2$, it is **not significantly high** based on the diagram. 8. **Calculate percentiles:** - To find percentile for value 12, count how many data points are less than 12: Values less than 12 are: 8,3,5,11,2,4,8,1 (8 values) Percentile rank = $\frac{8}{12} \times 100 = 66.6$ approx. 9. **Calculate 75th percentile (Q3):** - Sort data ascending: 1,2,3,4,5,8,8,11,12,15,17,22 - Position for 75th percentile: $P = 0.75 \times (n+1) = 0.75 \times 13 = 9.75$th value: interpolate between 9th (12) and 10th (15): $$Q_3 = 12 + 0.75 \times (15 - 12) = 12 + 2.25 = 14.25$$ - Note: User gave 13.5, but calculation shows 14.25; user’s data approximation likely used a different method. 10. **Calculate 25th percentile (Q1):** - Position $P = 0.25 \times 13 = 3.25$th value: between 3rd (3) and 4th (4): $$Q_1 = 3 + 0.25 \times (4 - 3) = 3 + 0.25 = 3.25$$ - User states 3.5, again slight difference due to method. 11. **Calculate 50th percentile (median):** - 6th and 7th values after sorting are 8 and 8. $$\text{Median} = \frac{8 + 8}{2} = 8$$ - Median equals computed 50th percentile, so **yes, median is 8**. 12. **Calculate interquartile range (IQR):** $$\text{IQR} = Q_3 - Q_1 = 14.25 - 3.25 = 11$$ - User’s value was 10; depends on percentile method. **Final answers summarized:** - Mean $\bar{x} = 9$ - Variance $s^2 \approx 43.09$ - Standard deviation $s \approx 6.56$ - Coefficient of Variation $\approx 72.89$ - Z-score for $x=22$ is $1.98$ (not significantly high) - Percentile for $x=12$ is approximately $66.6$% - 75th percentile $\approx 14.25$ - 25th percentile $\approx 3.25$ - Median (50th percentile) is $8$ (yes, it is the median) - IQR $\approx 11$