1. **Stating the problem:** We are given a frequency distribution table with cumulative percentages (percentile ranks) for intervals of $x$ values. We want to find the 90th percentile value $X$. 2. **Given data:** - Interval 20-24 corresponds to 60% cumulative percentage. - Interval 25-29 corresponds to 90% cumulative percentage. - Interval 30-34 corresponds to 100% cumulative percentage. 3. **Understanding percentiles:** The 90th percentile is the value below which 90% of the data falls. From the table, the 90th percentile lies in the interval 25-29 because the cumulative percentage reaches 90% at the upper bound of this interval. 4. **Formula for percentile within an interval:** $$ P_k = L + \left(\frac{k - F}{f}\right) \times w $$ where: - $P_k$ is the $k$th percentile value, - $L$ is the lower boundary of the interval containing the percentile, - $k$ is the percentile rank (here 90), - $F$ is the cumulative frequency before the interval, - $f$ is the frequency of the interval, - $w$ is the width of the interval. 5. **Calculate values:** - $L = 25$ (lower bound of 25-29) - $k = 90$ - $F = 60$ (cumulative percentage before 25-29 interval) - $f = 90 - 60 = 30$ (percentage within the interval) - $w = 29 - 25 = 4$ 6. **Apply formula:** $$ P_{90} = 25 + \left(\frac{90 - 60}{30}\right) \times 4 = 25 + \left(\frac{30}{30}\right) \times 4 = 25 + 1 \times 4 = 29 $$ 7. **Interpretation:** The 90th percentile value is $X = 29$. **Final answer:** $$X = 29$$