Probability & Statistics – Problem 1: Determine which statements are correct (multiple choice): A

A high school has a frequency histogram for the scores of 100 students, with class intervals $[50,60),[60,70),[70,80),[80,90),[90,100)$. The frequency/class-width (frequency density) values for the intervals are: $[50,60)$: $2a$; $[60,70)$: $0.04$; $[70,80)$: $0.03$; $[80,90)$: $0.02$; $[90,100)$: unknown (to be determined from the total). Determine which statements are correct (multiple choice):

A. $a=0.005$

B. The estimated mean score of the 100 students is $73$

C. The estimated 80th percentile of this dataset is $85$

D. If proportional stratified random sampling is used to select 5 students from strata $[70,80),[80,90)$, and then 1 student is randomly chosen from these 5, the probability that this student comes from $[80,90)$ is $\dfrac25$.

1) The total histogram area equals 1: $$10\big(2a+0.04+0.03+0.02\big)=1,$$ so $$a=0.005.$$ Therefore A is correct.

2) Use class midpoints to estimate the mean: $$\bar x\approx55\times0.05+65\times0.40+75\times0.30+85\times0.20+95\times0.05=73.$$ Therefore B is correct.

3) Cumulative frequency of the first three classes: $$(0.005+0.04+0.03)\times10=0.75.$$ Cumulative frequency of the first four classes: $$(0.005+0.04+0.03+0.02)\times10=0.95.$$ So the 80th percentile lies in $[80,90)$. Let it be $x$: $$0.02(x-80)=0.80-0.75=0.05,$$ thus $x=82.5\ne85$. Therefore C is false.

4) For strata $[70,80),[80,90)$, the frequency ratio is $0.03:0.02=3:2$. Proportional allocation among 5 students gives 3 and 2, then one is selected uniformly from these 5: $$P=\frac{2}{5}.$$ Therefore D is correct.

Hence the correct options are $A,B,D$.

Option A is correct because the total histogram area condition gives $a=0.005$. Option B is also correct, since the grouped-data mean computed from class midpoints is $73$.

Option C is incorrect: the estimated 80th percentile is $82.5$, not $85$. Option D is correct because proportional stratified sampling between the two intervals gives counts $3:2$, so the chance of selecting a student from $[80,90)$ among those 5 is $\frac25$.

Therefore, the correct option set is $A,B,D$.